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```Til K CALCULATION OK

CIIANCK-WHEKLS

SCREW-CUTTING ON LATHES

TUK CALCULATION OK

CHANCE-WHEELS

SCREW-CUTTING ON LATHES

TllK CALCULATION OF

CHANGE -WHEELS

1*OK

SCRIM-CUTTING ON LATJIKS

A PRACTICAL MANUAL

I tilt Illl- USl> til-

MANUFACTURERS, STUDKNTS AND LATH1CMKN

1), Dh VRIKS

WIIII 4(1 ILLUSrHAriONf)

K, & F, N, Si'ON, LIMITED, g; 1IAYMAUICKT

\$eto fortt
Sl'ONT *Sr C1IAM11KRLAIN, 123 r.IHKRTY STRKKT

PR Kl< ACM

IP . a i unoiis < \f( uni'tirtm e lh.it the calculation ol (hani>,e-
\\he<li foi jlir tuttin*; t>( diffeient pitches ol thiead on
a lathe, hmu'vet simple -.itch ,i udt ulaliojp may he, is
u>jnp.itativel\ but little Known, hc-int;, lor the tn.ijoii'ty ol
Ihosv most ( loscly intctrstal in ttu* suhjc'ct, shioiuletl HI
in>stt'i v

Many \\hosr thcoK'lual knowl'<ljt' is quite 1 suliuicnl
to jMKihlt tht'tn lo face the problem, have had so little
jnaetual expenenre in setew-uitttng th.it tliey .u'e uiiiiblc
to to <kej)ly into Ihe m.ittet, and pie.sent, in a eleui (in<l
siiiiplr inatHH'i, the tlifietenl variations whiuh in.iy possibly
occur.

The \\n ,tlei nutubei ot ineehanics, even the ytuui;>ei ones,
pu.-.tss (no sh;hl a the(*retltil knowledge to permit ol then
btuhlui", up a -.ystein by themselves.

thru aie, ol ionise, iiHHhaiuch who are quit< j capable
o{ uoikui^', tnit tlic iH'fu.ssiuy rukulation, but so many of
them I speak ftom petsonal experience regard their
knowl<*d!' ( e an aunt* or less of n secret, and uy, ut any rale
to theniM'lvi"., "Why should 1 impart to others what ha*',
taken tut 4 M ituu'h trouble and cost me so much money
lo Irani j 1 *'

The pin pose ol the present treatise is to unable any
one, who Is ptepared to take tine trouble to study it carefully
U Irani how to *ah ulate change- wheels piopeily,

vi The Calculation oj Change- Wheel*.

I have deemed it expedient, foi the sake of those of
my readers who have but a supeificial knowledge oi tlu-
lathe, to give a shoit desenption of this tool, in so fat as it
is connected with sciew-cuttmg, to winch 1 have atUeil a
de&ciiption of the various types of thiuad to be met with,
with the necessaiy tables appended, as also a nunihet ui
piactical hints, with icfcience to anew-cutting, together with
the operations connected theiewith.

I have puiposcly refiained from including, 1 ; a munhei (
tables giving the change-wheels requited for the v.uiuun
pitches of thicads on different lathen, in place of whuh a
large numbci of piactical examples aie given which cuvei
eveiy possible variation likely to be met with in practical
work. Experience has taught me that the inclusitm o! such
tables only leads to purely mechanical work demanding
no effort of the mind, whereas, in each partieuhu c.ue, tlu
consideration should be given to the special work in ham!,
so that in cases of exceptional difficulty, whore one is obliged
to set to woik without the assistance of such tables, the
manner of calculation may not be unfamiliar,

It is my earnest wish that the present work may prow
useful not only to students, but also to those engaged in
practical work,

I), DK VRIKS.

HILARY

*

CONTKNTS

CUAITKK 1.

I'AGK

THF LA mi , , , .. , y

CHAPTER 11,

{/*) Systems ,, ,, , ,, .. no

(/*) Whflt C "ha nt>e Wheels aic to he found on a Lathe . 15

{/) Thf Cutting ol Metrie Thieuds on a Lathe with Motile

Leadheiew , ,. , . . , 16

(/} The Cutting of Mni'ltJih Threw Is on a Lathe with

iCfij'hhh l,e,ld,seiew .. . . ,, ,, 3O
if) To C*ttt Kiifjltsh Thread?, on u I'ilh< i with Metric

(/) Tin C'utiini 1 , ol Metric Thiends i>n a Lathe with

tyiglihh Lesidwttnv ,, ,, , 37

(,C) Thr Wheel with is? Teeth ,, ., , . ,, 30

{//} Method foi Caieulnting Appioximalo {''tnefions ., 32

{/) The I 'tool of the Sum , ,, , ,, ,. 51

{&) i\ nm up the Wheels , .. , 53

{/) Thrfc'uftmj, with Double Compound Train 5/1

(i) TJu? Cutting oi* Left-hand Threads ,, ,, ,, 55

vin The Calculation of Change- Wheels

CHAPTER lit.

THREADS AND THEIR CONST. RUCTION ,, Af ,,

(c) Screw-Cutting Tools 65

((?) The Cutting of Double and Multiple Thi ended Sacws 73

(/) The Cutting of very Coaise Thiead 75

(g) The Hendey-Noilon System 77

THE CALCULATION OF

CHANGE-WHEELS

FOR SCREW-CUTTING ON LATHES

owri-r-

CHAPTER I

THE LATHE.

TliRKAHS, both internal and external, can be obtained in two
different ways, the simplest of which is to cut the thread by
means of taps, dies and chaseis. In the smaller sizes, the
are cut with dies, but in the larger sizes too much material has
to be removed. Tapping, however, is far moi e general than the
use of dies, as in most cases, external tin cads can be obtained
in anothei way, viz. : on the lathe, whilst internal threads
can only be obtained on the lathe at considerable expense,
M 01 cover, internal threads are to be found in a number ol
different places on the largei machine parts, and so it would
be well-nigh impossible to put these pieces on the lathe foi
the puipose of cutting the threads. On the othei hand, a
bolt 01 screw-spindle, as a rule, can be set on the lathe, and
thieads may be cut by means of a common tool. It is just
for this reason that, whilst a large number of i in. external
thieads are cut on the lathe, I in. threads in holes aic, with
but Few exceptions, cut exclusively by tapping. The practice,
however, of cutting internal threads of more than 2 in.
diameter on the lathe, whenever the work-piece allows it, is
becoming more and more general.

The object of the present work is to give a detailed
description of the way in which it is possible to cut the various

B

2 The Calculation of Change-Wheeh

thieads on the lathe, and thus to answer, as fully as possible,
the question " How are the change-wheels to be calculated
foi screw-cutting on the lathe ? "

In order that this woik may also be of service to those who
are not fully conversant with the lathe, the following points
will be treated successively, viz the gcneial constiuction of
the lathe, moie especially of those paits of the Lithe used in
screw-cutting , the theory of the calculation of change-wheels
and screw-cutting m practice

FIG. i

The lathe, as originally constructed, was not intended foi
sciew-cutting Fig i shows a lathe as it was fiist constructed.
On this lathe a rotary movement was imparted by means of
a driving belt to the headstock and workpicce only, all other
movements being executed by the operator himself.

Within a compaiatively short time, however, more wa.s
demanded of this machine, larger pieces were required to be
machined than was possible wuh direct belt drive, and the

for Screw-cuffing on Lathes,

3

double back t*vai was mtioduced , it was deshed to move the
tool on the mateua.1 automatically, and to obtain this, the
iest was mounted on a carriage and moved by means of a
leadscrew which motion was impaitod by means of eithet
a belt or a iiam of ^eais fiom the hcadstock. The intto-
duction of a train of gems on the apt on made it possible
not only to move the cairiage ovej the whole length of the
bed foi sliding, bul also to move tlie rest automatically
in a ttansverse dneetton over the eauiacie itself foi suifictm>.

wns <o arranged that by a set of gears of various diameters,
a variable, but at the same time foi each train of gcnn*
fixed ratio between the number of revolutions of the head-
stock, I.G, the workpiecc, and the loadacrow was obtainable,
thus making it possible to cut different pitches of threads on
the lathe. Fig, 2 given the general arrangement of such a
lathe. *

B 2

4 The Colt illation of Changc-irhceh

The leadscrew revolves m the leudseiew-nul, which is
fixed to the apion, and, as this nut cannot revolve, it travels
along the leadscrew, the caniage at the same time maKtn a
coi responding movement

The movement of the carnage aheady causes a ronstdeiahle
piessuic on the thread of the leadsciew and the nut, whit h
is still inci eased by the cutting of the tool on the matenal,
and, as a natural icsult, both the leads<iew and the nut aie
exposed to a ceitam amount oi wear. 'I his weai i\ ttnther
mci eased by swaif and chips falling on the le.ulserew, and their
getting between the nut and thiead,

It is evident, as far as the leadsciew is concerned, that
this wear will only affect that poition ovei which the mi!
travels on the thiead. As the woik on the Kit lie v,uies HI
length (but is as a inlc considerably shoitei than the m.iximwn
distance between the centies), the wear of the tluead is
gieatest on those parts of the leadsciew where the nut mv<*s (
and after being in use foi a certain time, it is impossible to
pi event the leadsciew being scarcely worn at all at the end
but considerably woi n in the centre, and worn most f nil
close to the headstock, The wear of the nut, however, is
fairly even.

The nut was foimeily made solid, consequently it was
impossible to icpair the wear. It wus soon seen, however,
that it was piefeiablc to have half nuts, .so that not only can
it now be repaired, but, by means of the lever a (Fig, 2), it
can also be opened and closed.

This has led to the attainment of a number of advantages
First and foiemost, the possibility of repairing the nut just
referred to A downward pressure of the level a keeps both
halves of the nut closed so as to grip the losukserew. The
two halves of the nut hb move in a vertical direction at the*
back of the piece c, and are provided with pins which fit in
cccentiic slots in the citcular plate which revolves on point /
Fig. 3 shows these cccentiic grooves in the plate, If the pins
of the half nuts aie shifted by moving the levi ,/, the half
nuts travel the double distance A B (Fig, 3), vk, : th upjnir
nut up and the lower one down, the half nuts being thus

jor Screw-cutting on Lathes

entirely disengaged from the thiead, causing the motion
imparted to the carnage by the Icadscrew to cease im-
mediately.

In the eailier types of consli action, with the solid nut, the
carnage had to be moved by hand by means of a handle
placed on a spindle in the apron, with a bevel gear on the
other side of the spindle to which this handle was attached ,
this in its turn meshed with anothei bevel geai fixed on the
hub of the nut. In this way the
nut was made to i evolve ovei the
leaclsciew and the cainage was
moved over the bed. But it took
far too long to move the cainage
any distance at all ovei the bed,
besides being very fatiguing work
The nut, being in halves, can no
longei revolve, but it can be
opened. A lack is to be found on
the bide of the bed in which a
pinion meshes to which motion is
impaited by the hand wheel h
(Fig, 2), by means of which the
caniage can be quickly disengaged
fiom the leadsciew, and a quick
and easy hand movement is secured.

enumeiated here have been clenved
from the split nut. One gieat
difficulty, however, still remains,

viz,, the diffeient weai on a ceitam length of the leadscrew If
this happens to be more worn in the middle than at the ends,
it is impossible to cut a true thicacl

Now, in comparison to the work oidmaiily pei formed on
a lathe, but little screw-cutting is done. The greater part of
the time the leadscrew is thus engaged foi the feed motion of
the caniage and for surfacing* For this reason, the movement
imparted to the carnage for screw-cutting, has been separated
from that for feed motion, A separate shaft, provided with a

FIG

6 The Calculation of Change- Wheels

for St.rew-cit,tting on Lathes.

7

keyway, impaits motion to the pinion which meshes with the
lack (Fig 4), by means of bevel and spui-geais The sliding
movement of the caniage being accomplished in this mannei,
the leadscrew is only used foi screw-cutting In still latei,and
principally Amencan constiuctions, the two shafts have finally
been united in one, the leadscrew being now piovicled with a
keyway, for sliding and sui facing the leadscicw simply acts
foi sciew-cuttmg, and so the same object is attained with one
shaft as is obtained in Fig 4 with two, viz., the thiead of the
leaclsciew is used for sciew-cutting only

FIG. 5

In Fig, 2 the gearing for the motion of the leaclsciew from
the head spindle is cleaily visible. Wheel I is keyed to the
head spindle ; icar wheels 2 and 3 mn loose on studs fastened
to the levei 4 By means of knob 8, this lever can be laised
to hole 5 or lowciecl to hole 6, If the lever is placed in
position 5, wheels 3 and I become engaged, and wheel 10 on
spindle 7 revolves by means of wheel 9, Wheel 2 now mns
to no pin pose. If the lever is placed in position 6, wheels
2 and i become engaged, and wheel 3 is brought into play by
means of wheel 2, thus causing wheel 3, as well ab wheel 9 and

8 The Calculation of Change-Wheels

spindle 7 to rotate in an opposite direction In the illusli.i-
tion the levei stands midway, so that wheel I engages neither
of the wheels 2 or 3, consequently, although the lathe spnulk*
lotates, the leadsciew is not rotating Wheels I, 2, 3 ,unl <j
have the same numbei of teeth, so that the wheels on spindle 7
make precisely the same numbei of revolutions as the lathe
spindle. Wheels 10, 1 1, 12 and 13 are the actual chan&e-wlu'c'ls,
and can be easily mounted, dismounted, 01 changed Wheels
ii and 12 rotate on a sleeve on spindle 14, and consequently
make the same number of i evolutions, so that wheel \?,
transmits veiy slowly to wheel 13 the motion hnpaih'd to
wheel ii In the illustration the gearing between wheel 9 to
the leadscrew is accomplished by 4 wheels wheels 10 and 12
being the driving wheels, ii and 13 those cluven It is evi-
dent that the motion of wheel 9 on spindle 7 it. mipaited
but veiy slowly to the leadsciew, in the same iatio as the

product of the number of teeth on wheels 10 and 12 to the
number of teeth on n and 13 Precisely the same is to be
seen in Fig 4. Wheel 13 can, however, be driven by means
of a wheel engaging both wheels 10 and 13, without the
mteimediate wheels ii and 12, thus solving a,s an idle wheel,
m which case wheel 10 is the chiving wheel and 13 the
one driven The iatio between the numbei of icvolutuws ol
the lathe-spindle and leadsciew is identical with the nttm
between the numbei of teeth on wheels 10 and 13.

Wheels n and 12 are mounted on a sleeve iimnint' mi
stud 14 (See Fig. 6)

This stud must be movable in accoulauco with the
dimensions of the wheels, and is consequently placed in
casting called the sheai 01 swingplatc at the end of the lathe
This shear (Fig. 7), has two long slots, so that the stud can
either be brought close to the leadscrew B, for small wheels,

for Screw-tutting on Lathes. 9

or moie to the icai foi latgei wheels, at will In ordei to pei-
mil of working with five or six wheels, <i second slot is to be
found in the sheai This shcai turns on the le.adgc.rew B,
and is held in position by means of the two bolts to be' seen
in the cucular slots When the mtei mediate wheels have been
accuiately set in the wheel on the leadscrew, the shcai, which
was first loweied to its full extent, is laiscd till the mtci-
mediate wheel engages the uppei wheel piopcily, aftci which
the shear is fastened.

Fig, 5 shows an Amencan type of lathe, on which it is
not necessaiy to change the wheels for diffeient pitches of

FIG. 7

Lhieads By means of a conc-geat to be found under the
headstock and at the left-hand side of same, the latio of speed
between the lathe-spindle and the leadsciew can be varied
by the simple movement of a lever. The necessity of calcu-
lating the change-wheels is done away with, all that is icquiicd
being the placing of two levels in a ceitam position indicated
in the table. The manner in which this result is attained will
be {wilier described in Chapttte HI,

,* ]

io The Calculation of Change- Wheels,

CHAPTER II

THE CALCULATION OK CIIAN(2j!.-WHI''hLS

(a) System*

IN the calculation of change-wheels foi sciew-tuthng on the
lathe there is one difficulty, and that is, the diifeience between
the English and metric system of measuicmenLs. It is not
Insurmountable, but it does not icnclei the task any easiei,
and has been the cause of a consideiable amount of tiouble.

In the calculation of change-wheels it is <i mattei of in-
difference whether a light- 01 left-handed sctew is to be cut,
internal 01 external, 01, finally, the exact internal 01 external
diameter of the thread The one essential question to be
unit of length ?

For this puipose two units exist , ist, the inch ; 2nd, the
centimetre

Foi both these units of length the numbei of Kvolulions
of the thread are tcimed "numbei of thiedds."

The length of a single thread is spoken of as " pitch."

The number of thread* is thus determined dy the numbet of
revolutions per ttmt of length

If the pitch is indicated with the inch as the unit of length
we speak of " English thiead." If the pitch is indicated with
the centimetre as unit of length, it is culled a " metnc thiead "
Both, however, have a system, which rs further treated of in
Chapter III, but which, as such, has nothing a t all to do with
the calculation of the change-wheels. ,

If but one of these two units, cither the inch or the centi-
metie were exclusively adopted as the stunclau! unit, then the
difficulty refeired to at the beginning of this chaptei would

for Strew- cutting on Lathes 1 1

entirely disappeai. But the inch and the ccntimetie aie em-
ployed together , and not only that, but theie is also a lack of
unifoimity with legatd to the leadscrcw, one makei cutting
the leadscrew according to the English, and another accord-
ing to the metric system. English and American lathes
usually have a leadsci ew cut accoidmg to the English system ,
Ficnch and Swiss makers cut it almost exclusively accoidmg
to the metnc system, whilst Geiman manufacttueis employ
both systems, though the pieference is given to the English.
Foiu variations aie thus possible .

I. A metric thread to be cut on a lathe with metiic

; 2. An English thiead to be cut on a lathe with

3. An English thread to be cut on a lathe with

4 A metric thiead to be cut on a lathe with English

Buefly summarized

To cut i. Metnc on metiic

2 English on English

3 English on metric
4, Metric on English.

If one desnes, once and for all, to be able to calculate
the change-wheels loi eveiy vanety of pitch, it is impel alive
to know these four varieties thoioughly, as they can occur
intermingled.

1st Axiom, -The number of tlircadt, ts to be determined by
the pitdi of the leather ew and the uilio of the number of revo-
lutions of the laths spindle to that of (he Icadscn'io

This axiom holds good for all four UZ.WM.

The ratio of the number of revolutions of the lathe-spindle
to that of the load-screw is obtained by means of wheels
(change-wheels)

When the spindle of the lathe has completed one revolu-
tion, then the woik on the lathe will have also completed
one revolution.

1 2 The Calculation of Change- Wheels

If the number of i evolutions of the lathe-spindle <uul lead-
sciew rue the same, so that the leadsciew has also completed
one revolution, then the camai>e has moved a distance dining
If a tool has been placed in the toolholdei, so that it can cut
the woik-piecc, then piecisely the ime pitch will have been
cut on the woik-picce as that on the leadsciew. With an
equal number of revolutions of the lathe-spindle and the lead-
screw, the tin cad cut on the work-piece will have the same

If the lathe-spindle has completed one full i evolution,
but the leadsciew on the other hand only half a i evolution,
then the carnage, and with it the tool, will have moved in a
straight line ovei a length equal to half a pitch of the
leadsciew It is thus only when the lathe-sptndle has
one full i evolution, two threads aie now to be found
on the work-piece over a length equal to one pitch of the
leadscrew. The ratio of the numbei of i evolutions of the
spindle to that of the leadscrew was 2 I , the latio of the
number of thieads per unit of length of the woik-piece to
that of the leadscrew was also 2 i. Hence it follows

2nd Axiom, The ratio of the number oj i evolutions of the
lathe-spindle to that of the leadscrew u the same as the pro-
portion of the pitch of the thread to be cut to that oj the lead-
it crew

Axiom 2 is also applicable to all foin cases.

Foi example, the leadscrew of a lathe has a pitch of one
thread to the inch It 14, lequiied to cut two threads to the
inch. The piopoition of the pitch to be cut to that of the
leadscrew is thus 2 i Accoiding to axiom 2 the ratio
of the number of revolutions of the lathe-spindle to that of
the leadscrew must also be 2 I

The leadscrew has thus to complete one i evolution to
two of the lathe-spindle. The leadscrew leceives its motion
fiom the lathe-spindle, so that the rotation of the leadsciuw
must be ictarded accoidmgly. The lotation of the lathe-
spindle is transmitted to the leadsciew by wheels. The pro-

for Screw-cutting on Lathes. 13

portion of the numbei of teeth on wheel 10 (see Fig. 2), to
those on wheel 13 on the leadscrew must thus be in inveise
piopoition to the latio between the number of revolutions of
the lathe-spindle and the leadscrew, which, in the example
given, must be 2 1 , the latio of the wheels 10 and 13 thus
becomes I 2. If then a wheel with 50 teeth be on the
sleeve of spindle 7, and one with 100 teeth on the Icadsciew,
with any desued idle wheel, a sciew of 2 tin cads to the inch
or }-mch pitch will be obtained on the work-piece with a
Icadsciew having one-inch pitch. From this we amve at
what is again applicable to all foui cases

jrd Axiom The- proportion of the number of the threads to
he cut to those in the leadscrew is in inverse ? atio to the pro-
portion of the number of teeth on the wheel on the lathe-spindle
to the number of teeth on the wheel of the lead-screw, 01 In
fiactional foim

Number of thicacls to be cut _

No. of teeth on the leadscrew wheel
No. of teeth on the lathe-spindle wheel

In this manner the calculation of the change-wheels foi
scicw-cuttmg is i educed to the woikmg out of a simple
fi action the number of threads to be cut being the
numerator, those in the leaclsciew being the denominator, or,
if it is denned to expiess the fraction in the same manner as
the wheels, i e. the numbei of teeth on the lathe-spindle wheel
on top as numeratoi, that of the Icadsciew underneath as
denominatoi, it is just the rcvcisc. The number of threads
in the leadscrew will then represent the value of the
numerator, those of the thread to be cut icpiesenting the
denominator As the pitch of the leadscrew on a ceitar
lathe is always the same, it follows that the value of *'
numeratoi is always constant,

We must here call especial attention to a rmsundcrsta
which so often occurs m connection with the question

14 The Calculation of Change-Wheels

the numeratoi or the denominate! A practical man can
generally tell fairly well which wheels have to be placed on
top and which underneath, but still, when the pitch of the

The screw may be denoted by the numbci of thieads pet
unit of length, m which case the numbei of thieads in the
leadscrew is the numerator of the fi action

The screw may also be denoted by the length of one pitch
of the screw , m this case the length of pitch of the scicw to
be cut will be the numeiatoi, the length of pitch of the lead-
screw being the denominator of the fiaction, the numeiatoi of
which will indicate the numbei of teeth on the lathe-spindle
wheel, the denominator indicating the number of teeth of the

Should the numbei of thieads of the sciew to be cut be a
multiple of those m the leadscrew, one is naturally inclined to
express it in number of thieads per unit , foi example, 4
threads per inch to be cut on a lathe with a leadscrew of
i thread per inch , should it not be a multiple, as for
example, each thread having a length of 7 mm., one is then
inclined to denote it by the pitch. If, m both instances, the
number of threads in the leadsciew be r pci inch, the fiaction
in the first instance will be

Number of threads to be cut wheel to be d liven

In the second instance, in which the pitch of the screw to
be cut must be 7 mm., the number of the threads to be cut

per unit is itself a fraction, viz - 5 4 , the fraction thus being

i 7

. 7 being the length in mm. of the pitch of the

fo _4 ^3 4

7 ~

screw to be cut, 25 4 the length in mm. of the pitch of the
lead-screw, so that, in this case, the length of pitch of the
screw to be cut can at once be placed m the numeiator foi
the driving wheel, the length of pitch of the leadscrew being

for Screw-cvitttng on Lathes 15

placed in the denommatoi foi the wheel to be duven In
actual calculation the fotegomg examples must be carefully
distinguished one from the othei

(b) What Change-wheeh aic to be found on a Lathe

This question piescnts itself each time change- wheels have
to be calculated, because the fraction which is formed by the
thiead to be cut and the leadsciew, must be changed into one
foimed from the wheels to be found on the lathe These
wheels should have such a number of teeth as will, within
ceitam limits, include the indivisible factors, viz 2, 3, 5, 7,
II, 13, 17, 19, 23, etc Some makeis supply these wheels in
a piogiession of 5, othei s with another progression. The
following set of change- wheels is, or should be piovided with
eveiy lathe, vu

15 = 3x5 60 = 2x2x3x5

20 = 2x2x5 65 = 5x13

25 - 5 x 5 75 = 3 x 5 x 5
30 = 2 x 3 x 5 85 = 5 x 17

35 = 5 x 7 95 = 5 x 19

40 = 2x2x2x5 100 = 2x2x5x5
45 = 3 X 3 X 5 105 = 3 x 5 X 7

50 = 2x5x5 115 = 5x23

55 = 5 x ii _ 125 = 5 x 5 x 5
01

If) = 2X2X2X2 42 = 2X3X7

18 = 2 x 3 x 3 44 SB 4 X ii

20 = 2x2x5 56 = 2x2x2x7

21 = 3x7 60 =2x2x3x5

22 = 2XH 66 = 2X3X11

26 = 2x13 78 = 2x3x13

28 = 2X2X7 88 = 2X2X2X11

34 = 2x17 96 = 2x2x2x2x2x3
38 = 2x19 1 08 = 2x2x3x3x3

One of the two foiegoing sets is gencially provided with

1 6 The Calculation of Change-Wheels

the lathe English lathes usually have a set of 22 wheels
some of which have the same number of teeth

It will be clear from what has been said, thus far, that
the easiest thiead to be cut on a lathe, i e the thread causing
the least tiouble m the calculation of the change-wheels, is
that having the same system as the leadsciew This will be
the case with the ist and 2nd cases refeired to on page II

(c) The Cutting of Metric Threads on a Lathe with Metric

Take the case of a lathe with a leadscrew having I cm.
(10 mm ) pitch.

It is requiied to cut 4 thieads per cm.

No of teeth on driving wheel _ No of thieads in the leadsuew
No~o7teeth"oia wheel to be driven ~~ No, of thieads to be cut

= 1 = _*L = gear-wheel 10 1 Sce F]g 2

4 100 = geai -wheel ou lead-screw J

It is required to cut 7 thieads per cm.

No of thieads in the leadscrew _ I _ 15 = driving wheel.
No of threads to be cut 7 105 = wheel to be duven

To cut i^ thread per cm.

No of threads m the leadscrew ,_ j_ _ 5 or 6 = cluvmg wheel.
~No of threads to be cut 15 75 go = wheel to be duven.

To cut 3 threads per cm

No of threads to be cut 3 90 = wheel to be driven

To cut 5 threads pei cm.

No of thieads to be cut 5 125 = wheel to be duven.

In the last example it is also possible to say, a pitch of
2 mm , in which case the fraction will be

Pitch in mm to be cut _ 2 _ ^25 = duving wheel

Pitch in mm of leadscrew lo 125 = wheel to be driven.

In both cases the result will natmally be the same.

Jor Scrcw-cnttuio on Lathes. 17

Pitch in mm to lie cut _ 7 _ 70 = duving wheel
I'Ucli in mm of leadseiew 10 100 = whcc>l to he duven

Pilch in mm to ho tut _ 5 5 _ S5 = dnvm^ wheel
Pilch in mm ol k'idseiew 10 100 ~- wluel to In duvcti

To cut a pitch of 7 mm

Pitch in mm to be cut

To cut a pitch of 5^ mm

Pilch in mm to ho tut
Pilch in mm ol k'idseiew

To cut 7 threads pei 22 mm

22

Denoted in pitch = a pitch of " mm

Pitch m mm to he cut _ "/- _ 22 ~ dnvinu; wheel
Pitch m mm ol leadsuew TO 7 " wheel to lie duven

,L

i

zot

Fu>. 8 Fiu. 9.

So fai it has always been possible to work with a single
tiam of wheels with any desued idle wheel. Fig, 8 shows a

single train.

1 8 The Calculation of Change-Wheels

In the set of wheels to be found on the Lithe, wheels with
cithei 22 or 70 teeth, as piesumed weie employed foi the
pieccding examples, weie not included A compound tiain
is now used.

22__2XH _ 20x55 dnvtntf \\liccls

70 ""7x10 ~ 35X HID wheels lo lie dnvcn

Fig 9 shows this comj)ound tiain

a and /; are the drivers, c and //those d) iwu The fixing
up of the wheels will thus be

55 x 20
100 x 35"

The wheels m the numeiatot, as well as those in the
denominate!, can be interchanged , a may thus be put in
place of b, 01 c in pLicc of </, or both may be c lumped , btit
interchanging of a clnvei wilh one to be diiven may nevoi
take place, as this would altei the value of the fraction and
an entirely different thiead would be obtained.

It is always advisable to tiy to get the smallest of the
drivers on the lathe-spmdle, and the laigcst to be driven on
the leadscrew, in oidci to obtain as lational a geating as
possible.

To cut 1 1 threads pei 14 mm I he pitch is thus 14/11 mm,

Solution

2X /

ro n x 10 j i x 10 55 x 100

To cut 3^ thread pci 40 mm. The pitch is thus 40/3 * 5
mm.

Solution 40/3" 5 = 40 ^ 4 X 10 ^ sox IOO (
10 3^x10 SX; 5^X35

To cut /] tin cads on 15 mm The pitch is thus 15/4 nun.
Solution

IS ^3X5 ^30x50 f
10 4 x 10 4x10 40 x ioo

Should the lathe have another pitch than 1 cm., this will
only necessitate a change m the constant of the leadsercw In
the fi action.

for Screiv-cutting on Lathes. 19

The following aie a few examples with solutions, dealing

i cm The pitch of the thiead to be cut in 16/9 mm
The pitch of the leadsciew is 5 mm.

c , , 1 6/9 2x8 20 X 40 . . , ,

Solution ' - = ^ = in case these wheels

5 9X5 45X50

, n 20x80

are too small

45 X 100

To cut a pitch of 3 mm. Pitch of leadsciew being- 7 5 mm.

2 -2Q

Solution 7 f 5 = ^

To cut 8 threads pei 13 mm Pitch of Icadscrcw, 7" 5 mm

Solution 13/5 '3 =2x6 5^20x65

75 8x75 8x7-5 80 x 75

In both the foicgomg examples, a wheel with 75 teeth
appeals among the wheels cliivcn, but is not included in the
specification given on page 15. With a leadsciew having a
pitch of 7' 5 mm. a wheel with 75 teeth will icpcatcdly
occui , in such a case the manufactiuci will be certain to
supply a wheel with 75 teeth.

To cut a pitch ot 20 mm Lead screw pitch 25 mm

c . 20 100

Solution 25 = 125'

t , . , 2O/3 2O 2 X 10 40 X 50

solution f.^i ~ ~

solution. 25 - 3 x 35 - 3 x as - fo x I2S -

To cut a pitch of 37-5 mm. Leadsciew pitch 25 mm.

Solution. 37-5^15X25 30x100
on. 25 10 X 25 40 X 50

To cut a pitch of 76 mm. Leadscrew pitch 25 mm.
76 4 x 19 40 x 95 80 x 95

Solution _

35 2'SXIO 25X50 25X100

C 2

2O The Calculation of Change- Whech

(d] The Cutting of English Threads on a Lat/ii unfit

In principle, this second case lescmblcs the fiist The
system of the leadsciew and the Ihicad to be cut is the
same.

Most lathes have a leadsciew with ^ in pitch, thus 2
pitch, the smallei sizes in, or 4 thicacls per Inch, whilst in
exceptional cases 2\ thieads pei inch aie to be found. Given
a ceitain pitch, the ft action can then be clctci mined without
any difficulty

Should the screw be denoted in a ceitain numbei of
sciew is placed in the numerator, the nunibci of thieads pei
inch to be cut in the dcnorninatoi Should the sciew be
denoted in the length of the pitch, then the length m inches
of the pitch to be cut is placed in the numeiator, the length m
inches of the pitch of the leadsciew being placed in the
denominator.

In practice the majority of threads aie cut accoi cling to the
Whitwoith system (see page 57), for which reason we shall
fust of all give a number of problems with solutions fot this

No of thieads to be cut pei inch

= 2 = _?! *_L _ _ 2 1 * S
16 16 x 12-5 ~~ 80 x 125'

Solution 2 = - 2 -*_L 2 .X. 40

7 3*5X2 35 x 80

for Screw-t,nttmg on Lather. 21

c . 4 2 2 X IO 40 X 50

Solution: = = J

45 5X9 45 X 100

c , 2 40

bolution : = - .

3'5 7

per inch I \ in, Whitwoith thiead = 7 tin cads pei inch.

Solution =

7 70

,, , t 2\ 2S 20 X JO

Solution - = = * .

ir no 5 5 x 80

inch.

c , . 2 2X4 2 X 'I 20 X ()0

Solution - r =- = ' =

2 ! 9 3X3 3^ X 15

To cut I thiead pei inch (not it 1 indi pih //), Leatlsciew

c , ., 2 2 X H 80 .10 X tOO

Solution w sss s=s ss.

H 7 35 25 x 70

Solution; 3 i aa ]ro sa| 50 i

2% n 55

In the following examples, the length of pitch is given.
The pitch of the leadsciew will consequently appeal in the
denominator.

22 The Calculation oj Change- IV liceL

To cut a j in. pitch Leadscicw 2 thicads pei inch =t
^ in. pitch.

Solution I^H^fo'

1o cut a | in. pitch. Lcadbcrevv 2^ tin cads pel inch

a= r in. pitch
2\

c . . }-?, }f. x 2-V it; x i\ 75 x icx)
Solution -- = ' (> = > ~ = x J ., .

i i 16 x i 40 x 80

2&

inch. The pitch to be cut = in. The leadsciew

ly

pitch ~ in
^^

ii'5

Solution -I2_= !I 5x2.5^115x1125.

_ 19 95 x 100

2^5

To cut a pitch of 4 % in. Leadsciew I pitch per inch.

Solution: l5 f = 3 X 13 = 6 5 X 75.
i 8 2x4 25 x 40

To cut a i J in pitch Leadsciew 2 thicads pci inch.
Solution il = I 3X2 == 2x I 3 _65 t

i- 32 2 X 1C 80

h The pitch

inch The pitch to be cut = ^ in The pitch of the

d La

Solution JL _ 5if! X 2& ^ 95 x_2/5 125 v 95
_i 9 9 X 16 80 x 90 "

21

In the foregoing examples practically every case which Is
likely to occur, has been treated.

for Screw-cutting on Lathes. 23

(a) To cut English Threads on a Lathe with

In the first and second cases considered, the system of
the thread to be cut and that of the leadsciew weie identical,
viz., in the fust case accoidmg to metnc measurement, in the
second, according to the English measuiement

In the thud case, howevei, the system of the thread to be
is divided per cm = 10 mm., or some pait 01 multiple
theieof, the screw to be cut being divided pei inch = 25 4
mm., 01 some part 01 multiple theieof

In the third case to be considered, this numbei 25-4 will
consequently appeal icgulaily eithei in tne numciatoi 01
the dcnommatoi, and will mvauably produce a fi action
which, with one exception, cannot be lesolved into whole
numbers

An equivalent must theiefore be found, by means of which
it will be possible to foim a divisible numbei from the nume-
rator and denominator of the fraction

This equivalent is to be found as follows 6^ in. =
16-509675 cm., taking for gi anted that 6\ in. = 16-5 cm.,
there is then a disci epancy of 0-09675 mm. per 165 mm.
of length, 01 rathei less than 0-06 pei cent., a diffeience of
piactically no mipoi Lance whatcvei.

If the number of tin cads to be cut be expressed in a
certain number pei 6 5 in , and the number of threads of the
leadsciew be also expiessed in a certain numbei per 6-5 in.
or 10 '5 cm., the result will be an equivalent which can be

per unit of length, in this case, 6-5 in. 01 16-5 cm., the numbei s
number of thieads to be cut in the denominator.

The following comparison can thus be formulated

No. of threads to be cut per 6-5 in, wheels to be driven

24 The Calculation oj Chaugc-lVhccL

variable for the same lathe, the numcuitoi is consequently a
constant factor foi a ceitain lathe,

Should the leadsciew have a I cm pitch, the hvulsciew
will then have 16-5 thieads pci 16 5 cm , and the constant
factor of the numciatoi will be 16*5, whilst, at the same time,
6* 5 is to be found as a constant factoi in the denominatoi,
and must constantly be multiplied by the number which
expi esses the number of threads to be cut pei inch I( both
these constant factois be multiplied by 10, the numbei 105
will always appeal in the numciatoi and the numbei 05 m
the denominator, in this way

constant factoi of numerator 165

denominator No of thieads to be cut pei in X 65

or .. ll x J 5

The equivalent is now complete , by icplncnu* threads pet
znckm the denominatoi by the actual numbei, a fi action is
obtained which will permit of the calculation of the wheels,

In the examples which follow, eveiy possible vaiiation
has been carefully worked out, fiom the simplest to the most
intiicate

Numeiatoi = _ 0x15 11x15

Denominator = No of thieads pei inch x 65 ~" 0x65

= 55X75 = 50X55
150x65 65x100*

Solution IIXI 5 = 11x75^55x75
4x65 20x65 65x100'

To cut 2 7 | tin cads pei inch Leadsciew 10 mm, pitch.

Solution IIX 15 = 4Xnxi5 = 2X2X 3x5x11
4X65 9x65 3X3X5X13

-, I2X 55 ^ 55x6o
45X13 45x65*

for Screw-cutting on Lat/tes. 25

To cut 5\ tin cads per inch Lcaclsciew 10 mm pitch.

c 1 4 11x15 1 1 X 30 30

Solution ; r = ~. = ,

5 5x65 11x65 65

bcicw 5 mm pitch.

In this case the Icadsciew has 2 tin cads per cm Conse-
quently for this p<irticulai lathe, the numeiatoi ib 2 x 165
= 330 01 II X30

Solution 11X30^60X55

8 X 65 65 X So

To cut } in yat, thieatl = I4thiea ( ds per inch Lcadscrcw
5 mm. pitch

014. ii xso 30 X 55

Solution ~ = , J .

14x65 65x70

To cut -]_ in. Whitwoith -thi cad = 20 tin cads per inch.

c 1 4* II X3O 30X55

Solution J = ; " .

20 x 15 65 X 100

Solution " ' SAA " t3 = " ; i5 =-7 V 5 =r 5 -
11x05 65 6x65 05

To cut 36 tin cuds pei inch. Leadscrew 4 mm pitch.
No. of threads in the leadsciew per cm , l t 01 2*5.

Solutmn. -5xnxi5 1 .nx-5_ 35X55.

36X05 I ' X' 3 ' T X 120

IT x 120 _ 55><i7o

(""* -I . JU J /\ IT ^' "n * * <\ *

boluuon. == 7x( . 5 _ 7xfl5 - J5X65

26 The Calculation of Change-Wheels

To cut a in pitch Leadsciew 10 mm pitch. No. of

threads per inch = = .

77

8 /

Solution * = ' -* = ^

8 - 8x65 65x80

7

per cm \

c 1 4. V x ii x 15 10 x ii x 5 55 x 100
Solution . (> , . -*= 3 = i- 3

J x 65 3 x 65 30 x 65

per cm \ Q .

Solution- Vxiixis^ioxii Xi S

a 7 b X 65 36 X 65

_ 5^x n _ 50 x 55
6 x 13 60 x 65"

To cut 9-5 tin edd per 8 inch. Leadscrew, 10 mm. pitch.
No of thieads pei inch, ^ 5,

- TI X 15 ^ 8 X ii x 15 _ no X 120

Solution-

To cut 25 thieads pei 3^ in. Leadscrew, 5 mm. pitch. No. of

3i 15
per cm, = 2

Solution 2 x u x I S = ^ X ii X 15 X 15
Vy x 65 ioo x 05

= 5S x (JO .
65 x 100*

for Screw-tutting on Lathes. 27

To cut a 2J> in pilch. Leadsciew, 10 mm. pitch No. of

I 2

. , A . 11x15 5x11x15 1 10 x 75
Solution -, , w = , = /

^ X 05 2 x 65 20 x 65

To cut 2 tin cads pet 6,1 in. Leadhtiew, 25 mm pitch

I 2

sciew pei cm,, ^f~ *

" i 5

t , . ^xiixi 1 ; 2xi>xnxi5 5sx 60

Solution ' , . J = -

i x ot; 1 X 5 x 05 k |u x 2t;

(/) 77^' Cutting (>J Mt'tm Tluead^ on <i Lathe witk

To some extent the fouith case lesembles the thud The
piopoition 10:25*4 also holds good, though with an opposite
meaning

Use is also made in this instance of the fact that 6" 5 in. is
equivalent to 16*5 cin

Suppose, for example, that the leadscrew has a t inch
pitch and 10 tin cads per cm. have to be cut, i.e. a I mm.
pitch, then, when the leadscrew lias completed 6-5 revolu-
tions, the lathe spindle should have made 165 i evolutions,
which can be formulated

No. of threads in the leadscrew pci 6-5 in, _ 6* 5
No, of threads to be cut pet 165 mm. ~~ 105*

The numerator of the fraction will thus, for a yivcn lathe,
always be equivalent to the number of threads per Inch in
the leadscrew X the factor 6*5; the denominatoi being
equivalent to a fraction, the numeratoi of which is the factor
165, and the length in mm. of the thread to be cut, the
denominator.

28 The Calculation of Change-Wheel \

Foi example, a 2 mm. pilch is to be cut on a lath
the numeiator will be 2x6 5 = 13

and the denominator will be

2

Foi this paiticulai lathe the numeiator will always be 13
The fust lesolvent of the fraction is a whole mmibc
obtained fiom the clenominatoi by placing the denominate)
of the fi action, winch is the denominator of the compouru

2X1^?

fraction in the numeiator, thus r

No useful puipose, howevei, is effected by this alteratioi
eveiy time The pitch of the thread to be cut is according!)
placed directly in the numeratoi, the fraction then benu
definitely foimulated as follows

Numeratoi = Pitch m mm, of thiead to be cut x No. of

Denominator = 165

Attention must here be directed to the fact that whenevci
the length of the thread to be cut is a fraction, it must nevei
be lesolvcd into a decimal, but must always be placed in the
numerator as a vulgai fraction, so that compound fraction 1 -
maybe icsolvable from numeiatoi and denominator by multi-
plication of both.

The following examples, from the simplest to the most
complicated, will make cleai what has been stated above .

pei inch

To be cut a 2 mm pitch.

Solution. 2X?X6- S= 2x13 = 20x65 ,
165 ii x 15 75 x no

To cut a 3 5 mm. pitch. Leadscrew 2 threads pei inch

Solution. 3'\$xr 3 = 35X6 5>

11x15 75x110

for Screw-cutting on Lafhc\. 29

pci inch ,

To be cut a ^ mm pitch

Solution ^ X1 3 = 10x13 = 10x1^20x05
11x15 3x11x15 11x15 ,15x110

To cut a screw of 8 thicads per I i mm Lc.ulsetcw n

To be cut a y mm pitch

c , . V x 13 13 20 x 05

Solution - h = = J .

11X15 8x15 J GO X \ "M

To cut a sciew of 5 thieads pei 18 mm Le.ulsdew >

c i 4 M^xis 13x18 6x13 s <! xfK

solution __ - - = r- ~~ * "

11x15 5x11x15 nx^ 55x115

To cut ,i sciew of 4 thicads pci 7 mm Lisulsucw ?\

Solution 1X2^X6^ 7x13 35X^5 .

11X15 4X2X6X11 I 10 XI 30

To cut a 7^ mm pitch Lcadsciew 2\ tin ends p<jt inch.

Solution , 7 ^ x 2 l*fia = 5 X J 3 ^ 5<> X 65
11x15 tixS 5 5 x tSo

To cut <i io| mm pitch. Lcadsciew i thuNid pci inch,

Solution 10 * x(5 i 2IXI 3 ^ 7XI3^3SX65

LIX15 4XI1XI5 11X30 55x100'

To cut a 42 mm. pitch Leadncrew i inch pilch.

Solution 42X6-5 = 42x13 ^ 7X rj ^ 70x65
11x15 2x11x15 sx n ~ 50x55'

To cut a sciew of 13 thieads pet 5 mm, Lcadscnnv /j

Solution, x/ix 2XS ^ 20x25

11x15 7-SX22 75x110'

Illlllllllllllllllllll

30 The Calculation of Change- 1 7 '//<'<?/ r ^

(g) 7/# Wheel with i 7 /re//'

In addition to the equivalent 6* 5 in. i<VS < m,, nhirh
has been employed in the thud <uid fourth cases, theie
is still another way of cutting English thu-.ul on a lathe
with metnc leaclsciew, 01 vice vet&d, winch is, by nicking
use of a wheel with 1 27 teeth

The pioportion between the cm. and the inch of 10 2}'4
can be resolved into one of 50 : 127

127 is not divisible fuilhei, <ind so, if a wheel with
127 teeth be employed, this i.ictoi can be- placed eilhej in
the numcratoi 01 the denominate! .

The thncl and fouith cases will then leseinble the fiint,
seeing that it is now possible to expiess the Knghsh tlmsul
in mm, whcthei it be the thieads in the leaclscivw r the
thieads in the sciew to be cut The ft action will thus be

Numeratoi = Pitch to be cut in mm
Denominator = fitch of leadsciew in mm

Denommatoi = No. of threads to be cut pei cm X 2*5-1

The following examples will cleat ly indicate* what Is
meant

Numeiator = 2 _ 20
Denommatoi = 12 7 127

The foregoing example, when wotked out as per the UiHt
comparison, will yield the same result, seeing* that : -

2 mm. = 5 threads pci cm.

Numeiator = 2 __ 3 __ 20
Denommatoi = 5x2' 54 ~* 12-7 "~ 127

for Screw-cutting on Lathes. 31

Solution - -- -= ~ 2 ___ = 4X50 = _40>5o_

3x254 6x1-27 6x127 60x127
01, accoidmg to first comparison,

i pitch = 10 mm.

Numerator = ip j; O _ 40x50

Denommatoi = 12 7 ~\$xi2 7 Y~~6ox 127'

inch

44

Solution -L_ = _J4_ = 40 x 55
12 7 7x 12-7 35 x 127

Solution -2-.= 9X25 = 45Xi2 S .
25 '4 254 50x127

inch.

Solution. JL= 45X4^45X50
25-4 28x25-4 70x127

4

screw 10 mm Pitch to be cut = ^4. mm.

8

25-4

Solution: _i_ = 25 '4_ = 20x127,
10 8xio 10x100

screw, the wheel with 1 27 teeth is always to be found amongst
the wheels driven, whilst, when cutting English thiead on a
lathe with metric leadscrcw, it is found among the drivers.

c , . 25-4 20x127

Solution ; - -- J ~ =

3*5Xio 35x100

32 TJic Cal( illation of Chtuigc-U7iec/\

Yci cut 4 in, <^as thicnd = 11 tlueads pei inch. Leadsnew
14 > nun. pitch

e i , 25 4 20 x 1:7

Solution J h = '

I I X 1C) IOC) X I IO

Io cut 3 tlueads pei 8', ui =- nu h pil<h Lcads(io\v

o ',

|l > nun. pitch

25-4
3

r o 3x10 2 5 x ' 10

To cut <) tlueads pci 1 1 in Le.ulsuew 25 nun pitch,

25 A
1 1

Solution 9__ ^0x25-4 = 45 xi -7

25 11X25 55x125'

"^^^C'/l ^ O V I ^ 7

Solution ^ "- 1 ^ = "

7X7 35X70

c i < 9x215-^ /is x 127

Solution = 1J , 7

2/1 X 5 50 X 60

(//) Melltod for Calculating Approximate F

Bcfoi e conimcncinjy wit IT, the actual calculation, the question
was pi opoundcd nuclei heading (//) on p<i t i>e 15 . " What chanp.e-
whools ue to be found on <i lathe?" This was indeed im~
ju'tMtive, as the change-wheels actually pie.sent on the lathe
have invuiubly to be taken into account, first of all because
the fraction must be lesolved into munbcis coucspondinjjf to
the change-wheels, and then, because the same faetois which
*u to make up the fraction must also be found in the change-
wheels. Should the fraction contain a fac toi not to be met
with In the change-wheels, then, aceoiding to the methods
now in vogue, a suitable set of wheels could not be iound,

for Sirew-cuttnio on Lathc\, 33

consequently, the thicad in question could not be cut without
obtaining one 01 mote \\hecls making up the lequisile faclois,
which, of coin so, would not be possible, as a certain thiead is
ijeneially tequired to be cut without notice, and theie is,
theiefore, no chance of eithei making 01* obtaining suitable
wheels,

Will siuh eases often ocem ? Not as a nile The
examples aheady ^iven cleaily show that even HI the case
of thieads which vuiy veiy consideiably, the wheels necessaty
foi cutting ,i line thiead can be found

In the set of change-wheels, j^iven on page 15, the following
fact 01 s weie found 2,3, 5,7, n, 13, 17, 19, 23; the factoi
33 was not met with in the second set, whilst on many lathes
UK far to is 17, i<), and 23 aie absent

If jetton ap/hw in //if fnution composed of the thread lo
fa' cut and the /twhcniv, wJiii/i cannot />e found ut the diau^e-
w/ft't'/s, then sitth n thififd tannot be ait accurately,

If it is absolutely necessaiy to cut such a thiead, a ft action
must be sought lot which appioaches the coirect fraction as
neaily as possible.

Lack of knowledge of the conect method of finding out a
I rarl inn apptoximalimj the tine fi action as closely as possible*,
loo often tesults in the calculation be-in}; skipped ovei, and a
li.ution btnn t t chosen which actually ^ives a thiead diffeiint;
rtnsiderably fioin the one tequired.

In addition, the fact ts loo often lost si^ht of that an
approximate fraction will still result in an uuseiviceahk thiead.

Support", for example, a fi action is found which yields a
tturad tlifiVrhtjLf only 0*05 mm, fiom the thread of the nut to
lit which the thread has to be cut. At first si^hl the differ
cmv appears trifling, but the en or which has been made is
ieally vwy tfj't'trt, so ^reat, indeed, that the thtead obtained is
wholly #,w'/d'.vx It must of course not be foi|,;otten that each
thread inct cases the error, which at the end of 20 tin ends will
result in a difference of 30 X 0-05 inm, = i mm Suppose,
fwther, that a threatl h.ts to be cut of 23 threads per inch,

1 2 OQ

the pitch being * 1 4 " tnm With a difference of

1 h 25 ',1 25^

I)

34 * The Calculation of Change-Wheels

0-05 mm pet thread, the diffeience at the end of 10 thicads
will be equivalent to one-half of the thiead, whilst at the end
of 23 thieads, the diffeience will amount to the en the thread

The foiegoing example clearly demonstrates that only
fractions diffenng by some thousandths of a millimctie, or
some ten thousandths of an inch, can be employed

How can such an appioximatc fraction be arrived at? 1

Regular piactice often enables one to find a fi action which
approaches veiy closely, without the assistance of any method

In one of his note- books the wntei found a fraction which
had been discovered, apait fiom any method, foi the cutting
of a 3 7 mm. thiead on a lathe with a leaclsciew having a
pitch of 10 mm

For this thread theie were no change-wheels, for a wheel
in which the factot 37 appears, which is indivisible, is not to
be found among an ordmaiy set of change-wheels

Foi this reason, according to the notes in question, the

77
fraction -~ was chosen, for which change-wheels could DC

f ,
found, since

7x11 35 X 55
*

-_- ---- -= - -

208 13 X 16 65 x 80

q 7 y?

Seeing that the difference between -~~ and ~ is simplv

10 208 L J

the difference between 3-7 and 3*701 = o ooi mm., so that
aftci 10 thieads the diffeience is still only croi mm., which
may be considered neai enough for all practical pin poses

Such gi oping about in the dark, however, is not at all
methodical, can take a veiy long time, and, finally, may not

The compound fraction, however, supplies us with a leacly
means of discovering a fi action which appioximatcs suffi-
ciently to pei nut the obtaining of what is piactically an

Suppose the fi action to consist of two numbcis, the
numciator and denominator of which arc both positive

for Strcw-tittting on Lat/ies 35

Let these numbois be lepiesented hy A and B, and A > B
This can then be tepicsentcd

^ =s rti 4- 'l /', < B 01 B > ;-!

Taking the icveisc of the last-named fi action, the i eduction
ran then be huthei continued,

K , ''' ^

= (h 4- , /j < ?i 01 i \> >'>

Continuing fuithei

which can be continued ad infimtum, and can thus be
expiessed

/-, " '-.'
in which

),,<>,< i <>' ''// i > >w

The (juotionts /?,, ^ 3 , ^. </*, ire termed indicators,
By substitution can be obtained

A

r
a i + ', etc., etc.

i
i

i
x

' - 1

1) 2

36 The Calculation of Changc-Wheeh

If ._ o, then the numbci of tenus is finite, in which
r*-i

A
case the fiaction is detcimmable, in that it can finally be

divided without leaving a icmaindci.

A
If the pioportion be mdctciminablc, and cannot con-

sequently be cxpiessecl by a fiaction with exactness, then
theic will be no end to the divisions, in which case the
number of teims of the compound fraction will be infinite

Every indeterminable numbci may be ief.aided as the Until
of an indefinite, non-iecumni> fiaction The /////// of a
lepcatmg decimal fiaction is a determinable ptopoition, ei>,
the limit of 0-3 is J.

To apply the foicgoing to a definite fiaction.
(i) Given A > B, for example

To expiess the fiaction as a compound Itnclinn

6961 6961

4- i

+ i

3 +

93 1

-f i

A + "

2'3
-h 1

6+
7

The indicators aic thus I, 2, 3, /j, 5, 6, 7.

for Screw-cutting on Lathes. 37

Consequently ^ 7 as a compound ft action =.

,

6+ *
7

2+ ----

* +
j \

4.4-

(2) Given A < B, foi example.

To express -^ as a compound ft action.

H3 _. _1_ _. i

355 355 , _i6

113 3 ~ l " 113
+ i

7+fg

A
If ..- <; I, the first mdicatoi can then be expiessed by o,

in which case the mdicatoi s will be

o, , } and T \ r ,

thus

113 i , ,.

- = o -f- as compound fraction

355 *

(3) Express the compound fraction 4 as an ordinary

fraction. 3

2

i

4

A = 4 + X == 4lt

3 + 2 + \ -3* S

38 The Calculation of C//ti/i^i'-ll iin A

(4) Expiess the compound haetion \ as an oidnuiy
fi action '

A
B

-f

3 +

= A + 1

T ~ (i

A 1 03

1? is thus =
B 215

The genual fonnula can now be expicsscd by putting
Icttcisin place of the figures given in the, foic;,Mmi;,> examples.

Given the compound daction a, determine the ouliiuuy
fraction b

A

given that

a i

b = 2

c = 3
</ = 4

a +

c<t

d

a />( d \ a b \ </ (/ i t
d but I b I d

bed -I- /; -f d
cd -I T

l Cd H- 1

</ " ;/

then

lu d s= 34
= 2
^/= 4
c d &= 12

bed =, 24
b 2

- the 1 nuuu'iator.

30 = the denominator.

so that in this ease the value of the fi action A '

B

Jor Sit'cw-in'titig OH Lathes 39

1^)1 any given value of a, l>, t, and d, the fi action can be
imim-diaU'ly detci mined from the ft action

a hi, d + a b -f a d 4- c, d + I
b t, d -f- b + </

To take the level se Given the otdinary fraction

(a b -f i ) 6 4- d
/>i+ i

dcU'imnu' the compound d.iction.

(a b + I ) ( 4- _. rf d <? + t -f a

/^ + 1 /><+!

= a -\ = indiCtilois.

*+'

c
'I'he indicators ate thus a, b, and c

(liven that in the foiegomg fi action the indicatois have
the following value a ~ 2, b 3, c = 7.

Then revcisnig the oidei of things in the foiegomg
example

(a b + t) c + a _ (2 X 3 -|- i) 4 + 2 _ 28 4- 2 = 30
bt -\- i 3x4+1 12-fi 13'

^O

The inclicutoih foi the h action ^ aie thus 2, 3, and 4,

The foregoing consequently pioves

(1) That uvcty detcrtiiinablc fraction may be expressed as
,i finite compound fraction.

(2) That every finite compound fi action maybe expiessed
us a (Interminable fi action.

Compound factions may be divided into:
(a) Symmetric.

SX

4<D The Cah ulation of Change- \l r hceh

If teims and compound fi action be cxpiesscd as

B = (^" ai a "~
indicators

then

A

is tcnned a symnictuc compound fi action because the in
dicatois end in the same oidei of sequence as they began , and

is teimcd a penodic compound fi action, bee , uuse the uulu atots
a,,, a it rtj occui peiiodiudly In both cases the ntunbei of
terms is infinite

77/# Findmg-ont of ApptMimating

Whcncvei the factois of a fi actual, act 01 ding to winch a
thread is leqimed to he cut, aie not rcpicsented by the
change-wheels belonging to the lathe, it is impossible, us bus
ah each/ been dcmonstiated above, to cut a theoretically
ft action, the value of which appioaches that of the teal
ft act ion so closely that the two may be icgardecl as piaetiealty
identical,

Such an approximating ft action can bu found by resolving
the fraction into a compound fraction, and toinunatlnp, this at
the second, thud, fouith, fifth, etc., indicator

Foi example

A . i

= ^ 4.

jor Scrctv-uiftmg on, Lather
Koi the fust quotient substitute

then the second quotient will be

l 'a ~~ , _ T

= rfi -+

CJa tf a

the thud quotient being

"'' ss a 4- ' = rfj (

?a 4- 4- i
j rt a 4- i

, etc. etc.

die the i educed uppioximatecl fuictions, the

values of which uie alternately ^icvitei and smallci than the

A A

value of , and they appioach moie and mote closely to ,,,

which may consequently be regarded as then limit

The ^i eater the ntimbei of indicatois, the smaliei the
difference between the approximating fi action and the exact
A

value of

B*

The following connection can be established between the

fractions and the indicators :-

p __ ji

a 4- i) H-i

3 ^a 4- PI

t on.suquently

P., =3 ,, P,, 4- PI and Qj = ., O a 4- Qi-

It follows, thcrefoie, as u geneial rule that
P, * a, i P n -i 4- P-a and Q = a n Q M ,1 4-
and this can be applied in the following mannoi :

4 2 The Calculation oj

(i) Given the fi action ^ Deteimme the compound
fraction, i.e. the indicators, and find an approximative fraction

A _ 51 , , i

=

B 16 ' ^ I a, 5

3 - 3

appioached
B Qi I ' *

A P a s X 5 4- I 1 6 , , i i , 11 i

._ = ,..- = 7 J = . limit appro.ichc'cl still tlosci

B g a 5 5

A P, 3(16) -1-3 ^i
- = ^ = , = ,,, the exact valiu'

B Oy 15 + 1 16'

Given the fraction

399

A = 3370 __ i

B 399 2 + I

i

i

7+6

p t

,,= , l '". 17

U j 2

{?,, = 4 < P a = , s (^^ 4- i) + <t, ~ 76 ( li - j + i ~- 9

17 y,-^= - ( , 2

i V

= 7 Pj = a 4 (,,( 1 tf a + 1) +!^ + 1) = 549 Q t , t (,.^ f J) \ a

a D = 6 ^ P r , = a R (4 ( fl 'i (^iss-t- + ii; t a + 0) + <*n (i<^ +

6 x 549 -I- 76

Ur, = n (4 (03 #a + i) + ^a) 4- rt^ + r

6 x 65 -i- 9 -- 399

The appioximating fractions are thus

3 j; 76 549 3370
i' 2 ' "9" ' 65 ' 399 '

Jor Screw tutting oit I,athc\ 43

(;) Deteunine the compound h action ami the appioxi-
matinj; h.utmiis <>( the mmibei 2 "j 182818 28459

A 718 \Si,S2.s i \$<) i

= ^ j 4 - -

I +

p, 2
(J, ~ I

- t P., ~- 2 X 1 + I - } Uu -* '

~> \> < v l > y / >sxiii- > "=r

1 ., : X { t >' - i VJ,i X I | I > ( ) i

' Pi I I

--- i P t - i x s | ^ n <Ji ~ i x H ' - ! ( > ~ I

P., '<)
= i P 6 ~ 1x114-8^-19 (](,=. i x .1 -M ~ 7 ( ) f) ~ 7

-()' ^' = K ' 7

(4J IMctminu the approximating ft actions for the niunbci

,

7 ,

7 +

,,-=? 15 P-.-333 (J.,= H)6

rt 4 t I'^SSS LU-^3

rt & 2tjJ P B 103993 (J B 33102

</ sa i

^ t =s j etc. etc,

<I H = 6

44 The Calculation o/ C/iange~ll'hccls

The approximating; fi actions aie, consequently,

3 22 333 ^55

i i x- j i Cl t I Hj.

i 7 i 06 113 33102
Fiom which the following can be detetmmed ~~

Axiom i The diffcience between two sm cessivc apptoxi-
matmg fi actions is, the signs not beiinj; taken into ronsideta-
lion, equal to the unit divided by the pioduct of its
or, in gcneial,

v = r "- 1> " + I = ( ~ J) "
* Q* ( J + , <J,(j + ,"

Should theic also be tlnee successive ap
h actions,

1> i) |>

1 n - i * n * u \ t

<}-/ Q/ <> + /

the fiist will then be gieatei than the second, the seunul
being smallei than the thud, etc

Example (sec page 42) .

A__ 3^70,
B - 399 '

the approximating fi actions are

8 17 70 5/19
i' 2' 9' 65

v _ -^ H-i - 1
; '~ 2 ' 18' 585

Axiom 2. The diffcience between the exact value of the

A
fi action B and one of the approximating fi actions will in-

vanably be less than the unit divided by the pioduet of thts
denommatois of this appioximaling A action and those follow-
ing, and also less than the unit divided by the square of the

/or S(rcw~tnfting on Lathe* /|5

denommaloi o( the fi act ion undei consideiation , 01, in

A _ P, ( - i r
A
( - i )* ( - i )"

<.>,.<], <V

A __ \\ i

u (] < (} ((_-!- o,, ,)

Application A __ 3370

H ^u()

I 1 , H P, _ 17 I', ___ 76 P 4

( ji = i ( j- ~ -' ( 2i u ( ]>

3V ~ l '7 <s '

3')y "^ - ! ~ 599 < ~" 2

30U " ' < t (H

137 () _ J 7 < !
399

< etc. etc

2 ^ 16

From which it follows th.it in order to obtain an appioxi
mating fraction, tliflenng only a millionth pail fiom the exact
vaku 1 , UK* (k'nomiiuttor must consist of at least /) figures.

The tliffcrcncen between two successive api>roxiniating
ft actions become ronliuually smaller, and are alternately
positive and negative. The difference appionches //, and
nmsequeotly the limit of the approximating fr.ution to the

r A

exact value of .,.

46 The Calculation o/ (. h<ui(~\\'hcci\

By mtcipolation anothei fraction can still be found behvcen
two appi oxi mating h actions
Gencial teim

P a P 4- V

1 '*>/ " H - T l * >t -

By taking in place of a, the values i, ', 3

P
"

_ t ), othci fi action s c<in he mleipolated between ' "" and

, both of which foi m an mcicasnu; 01 diminishing thaw,
as they both have the same sign

(i) Required, the interpolated fractions between ' = l ^

i ^4 549 ,- ,1 f , A 3370 .
and , . = > J of the fraction ,, = ' ( naie A M
O, 65 U 399 Vl ^ ' ;

an = 7 ^,~, = 6, 5, 4, 3, ? and i,

\\ <f tt l\- t + P_,
Q* ^iQ-i 4-o

l' = 549 O* =- f)5

P =17 ( ) = >

\\ = C> x 76 + 17 _ 473 i j . _ 3 X ;(. I- 17 \|5
() 6x9 + 2 50 () >t ~ 3 x 9 + i ;>9

P __ 5 x 76 + 17 _ 397 P,, _ n x 70 + I? ,( Mj

y 5X9 + 2 ^J7 O ff ""* : x () + > *" ;jo

r w = 4 X 7^ 4- 17 _ 321

Q 4x942 "" 38 (j,, i x 9 + > 1 1

The factions 93 *<*) ^45 3*1 W d7J

11* 20' 29' 38' 47* 50' u< lims " t-

twccn the fi.ictions 7 and y^, which nre anpioxiniathi"

2 (J " *

fractions of .- .
399

jor Strew-tutfuig on Lathe*. 47

P H

(2) Requned, the interpolated fractions between ' =

and . ' = of the same fraction

O ! 9

IV = <i,, 1V-1 + IV ~, IV ^3 x 17 f <^59

<j tf/^-.-f <J- U "3x24-1" 7

IV =- 76 <J = 9 }} ^ 2 x 17 -4- 8 = 42

() ? x 2 + i 5
IV- ,= i7 (],-,- 2

IV = i x 17-4- <s = 25

lV- 2 == U,-. = i <> 1x24-1 3

5 o 42 3 5
consequently, the appioximutm^ fiactions -;_, " " , he

between ' and

_,

/ 3 >*J

i 9

I /( )

(3) Rcquuccl, the interpolated Iraetions ' = and

l j r, _ 3370
UB 399 '

ft f) = 6 #,,_,; = 5, 4, 3, 2 and i

IV = 3370 U* 399

IV -i = 519 Q-i = f) 5

IV _ 5 x 549 + 7C _ ^2i IV 2 x 549 H- 7<^ _ "74

()/-"" ex <>K 4- o 3^4 ( ) 2 x 65 -j- 9 "~ 139

*^' J * J ' S <J*' r *" J ' * "

IV 4 x 549 + 7^ _ 2272 IV _ i x 5/19 4- 7^> _ r >-5

<J* ~ 4 x 65 + 9 "" 269 (J^ ~ i x 65 -H 9 " 74

IV _ 3 x 549 + 7^ = 1723
(j w , 3 x 05 4- 9 204

, . P . 625 1174 J723 2272 . 2*S21

the approximating fractions , , ' , ' ,uui
thus he between / and

48 The Calculation of Change- Wheels

Application Determine the compound fraction and the
approximating fiactions of the numbci 2*539954, so as to

5*4 12 ' 7
er piopoiio ~

r L

inch in cm

2*4 12 ' 7

obtain another piopoition as 01 ~ foi expressing the

r L 100 50 1 tS

A __ 2539954 = ^
B ~ io 6

3 +

*

24-

1+ '

14- '

2+ !
I I

The mdicatoi s aie consequently

2, i, i, 5. ! > 3> \$, 2, 3, 2, i, i, etc.
1235 28 33 127 1049 ?22 S
i i i 2 ii 13 50 413 876

The following and the approximating fiactions can he

obtained by interpolation between fractions 2l and ' 2 ^ -

n 50

P<J

127

Qo = 50

^ -

3

Pr, =

33

QB= 13

"o - \ ==

2 and i

TJ

1 ^

28

(J<i I I

P.

__a n

* W I "T" J W 2

2 x 33 4- 28

=r 9<l

Q*

a,,

Qx-, + 0,,-.,,""

2 X 13+11

~ 37

i x 33 4- 28

61

i x 13 4- n

24

By interpolation between s = ^3 (UK j P? _ 1049

Us 13 <Jv 413'

following can be obtained

for Sirew-tuttmg on Lathes, 49

IV _ a,, I 1 ,,, + \\, . = i X 127 + 3} = 160
<l a,, (} , 4-Q w . , i X 50+ 13 63

_ 2 x 127 + 33 __ 287

2 X 50+ 13 ~ H3

_, 3 x 12 7 + 33 = 4M

3 x 5 11 4- 13 163

_,} x 127 + 33 ^ 54i

4 x 50+ 13 213

__ S X 127 + 33 = ^8

5 x 5 + 1 3 -"63

__(> X 127 + 33 = 795
0x50+13 313

__ 7 X 127 -)- 33__ 022
7 x 50 + 13 ~~ 363

so that the following approximating fi actions can be found

\\ , ID jo 160 287 414 541 668 795

between ^ and ^ v,, fjj , n ^ ,63' 2,3-63' 3^3

and ^.

t's in Condition.

(i) It is icqunotl to cut 3- 1 , thieads pet 2}J, in Leacl-
icw ^ huh pitch.
Fitch to he cut ~' () Lcachcicw J inch pitch

Solution;
1

*' x 2 - ^^ ~ ' 13
x If - } ""x4"" 28

No wheel with ,13 teeth is to he found, and the numbci
f |3 ih indivisible. It will thus be neccssmy to find an approxi-

mating fraction,

'H I

fi action = ( ^ -- i 4- " j

1 +

5O The Calculation of Change-Wheel^

Indicators . i, i, i, 7, 2.

a, =

ff a = I P (J = I X 1 + I = 2 '

F, ^

tf a = I > P a = i X 2 + I = 3 O , = I X 1 + l = '2 ( )', ~ 2
ff/i = 7 P 4 = 7 x 3 -f 2 = 23 O,, = 7x2+1 = 15 o* - *,",

a rt = 2 P 6 = 2 X 23 -I- 3 = 49 Or, = 2 X i >; + 2 = 3 ' ( , 6 ~ ,
^ %,,,() .'

Intel polating between ^ and

1* rt w P_, -f P w n 3x23-1-3 26 ,

-" = " J = =- is obtained

() ^O"" 1 + O"~* 1 X 15 + 2 17

28

^ =i 5312 which is o 0045 l css tnun ^ lc Actual d.ietion

J ***

26 ,.

= i '5 2 94 0-0063

This cliffctcncc occuis in eveiy 2 thtcads, so that the
actual diffcicncc pei pitch is <inly o 00225

appiofichcs most closely to these two, so that the
wheels will consequently be

49 _. 7 X 7 _ 70 X 70^
32 4x8 40 x Ho*

(2) Requhed to cut a pitch of 3'7 mm, 1 ,cru Iscrcw

10 mm,

Solution : " .
100

Theie being no wheel with 37 teeth, and the number 37
being indivisible, an approximating' fraction will have to he
found.

for Screw-cutting on Lathes

Compound fiactum

IOO

4-

2 4-

1 f

2 4-

i -f

Imhcdtois ,ii( v thus o, 2, i, j, 2, i, 3.

-(!-,- 'Q,.,

!
-- ? , P. O X 2 4- I 1 Cj a = 2

! ., i

.- i i P, =- J X i 4- o = i ' O ; , = 1x24-1=3

Pi sr ^ y r 4- i j= '

x 4 "-" * f'x i ^r i ~"" .

<7< ~ 2

X 3 4- i =7

i x 7 4- 3 ~ 10

P 7 -= 3 x 10 4-7 = 37

O,, = 2 X 3 + 2 = 8

(J f) =2x84-3 = 19
y (! = i x 19 -f 8 = 27
Q 7 = 3 x 27 -I- 19 = 100

10

i\

Qi l

"O j

Q = 2
p t = I

Q.i 3

1^ J 5

4 ^^ O

Q * ~ s

T"^ *7

P fi ___ 10

Qe ~~ 2 7

100

The appioxiraating fmction ~ 3'74> which only chffcis

* J /

from the actual ft action by o*oo/{ mm. per tluead, may thus be
accepted ior <ill practical pui poses,

IO _ 2 X 5 _ 20 X 50

27 ~ 3 X 9 ~ 45 X 60"

(j } The Proof of the Smn.

The comparison thai 6*5 in, 165 mm., or an adopted
fraction, is not perfectly accurate. Should it be desired to
find out to what extent the fraction which has been arrived
at, and, consequently, the thread to be cut, deviate, this can

E 2

52 The Calculation of

be done, when a mctiic thiead h.is to be cut on a l.ilho having
an English Icaclscicw, by multiplying the numeiatoi ul the
fi action by the pitch of the leadsciew in mm The 1 pio
duct thus obtained should coincide with the' piodiut ol the
denommatoi of the fraction anil the pilch to In cut, t.e
numcratoi X pitch of leadsciew in mm. = denominatoi x pitch

Numeiatoi, denotmnatoi and leadsciew pitch lx nu known,
the pitch of the tin cad to be cut ran consequently be delei-
mined

On pay;e 28 the h action ^^ has In en detei mined !oi
a pitch to be cut of 2 mm, and a leadseiew ol 2 tlneads
pci inch

The piodutt of numeiatoi and loadsuew pile h m mm. is
thus 26 x 12-69975 01 26x 12 7 = 330*2. This piodmt when
divided by the denominate of the d action \\ill ;i\i the pitch
in mm to be cut with the* wheels deleunnu-d in, thus,
330*2 : 165 =2 ooi mm The pit< h is consequently cxutt
to within O'OOI mm.

7X J3 _ 91
11X20 ^20

is given on page 29 foi a pitch of io\ mm , with a lead screw
of I in. pitch

91 x 25-4 2311 'A

J J ^ J '

mm.

22O 320

The pitch is theiefoic exact to within iroofjj mm. Both
these differences may piactically be re^atded as of no tonse-
qucncc

In the case of a lathe having a metric lujulsciew on which
ICnglLsh. thiead is to be cut, the denominator should be
multiplied by 2 -54 The numeiator when divided by the
piocluct thus obtained, gives the pitch to be cut in inches

On page 24, the fraction for cutting f threads pei inch

with a leaclsciew of 10 mm. pitch is mven as l(> ^ ,

6x615

If the denominator be multiplied by 3*54, the* result will

be 1<5s - = l6s
6x65x2-5/1 990' 6"

for Screw-tutting on Lathes. 53

K.uh pitch cut is thus o- 1065656 in

The exact pitch = \ in. = cri6 in, so that the thiead
cut ihlfeis only by o-oootoio in

Note, that when cutting motitc thiead with an English
Icadsciew, the thiead cut is a fi action too coai se, whilst, on
sciew, the thread obtained is a fi action too hue.

(/&) Fixing vp the Wheels

It is not always possible to fix up the 4 wheels in the older
of sequence given in the examples.

FIG 10. FIG, ii

The following h action may, for example, occui .

125x55'

in which ca.se the wheels must be placed as per Fig. io
although the wheels 30 and 55 cannot mesh.

The fraction can, however, be ai ranged in another order of

54 The Calculation of Change-Wheels

sequence, viz > , which makes fixmer up possible (see
55 X 125 t. i i v

Fig. H), but care must be taken that the wheels of the
numerator are nevci placed in the denommatoi, 01 vice vci w?.
Should simple changing about of the factors in numeiator
and denommatoi, 01 one of them, be impossible, the fi action
is then resolved into the lowest possible factors, and .mother
combination of wheels sought for, which will give the same
pioportion between numciatoi and denommatoi, as, for
example

30x50 _ 2x2x3x5x5x5 _ 30x40 3OX.]o
55x125" 5x5x5x5x11 ""55x100 50x110

(/) Thread-t,uttiiig wit/i Double Compound Tram.

Should it be necessaiy to cut a tin cad consideiably coaiser

01 finer than that of the leadsciew, it can easily happen that
the necessaiy wheels aie lacking.

The fraction is * = IO x 1 5 A wheel with 10 teeth is
56 70x120

lacking 1 . If the numeialor and denominatoi ol the fi action
aic once again multiplied by 2, a wheel with i/jo teeth is
obtained m the denommatoi, which us also not at hand.

In such a case, the numcratoi and denommatoi of the
fi action aie icsolved into 3 factors, as, foi instance

2 = 20 = 2X2X S _, 20x25x30
56 560 5x8x14" 70x75x80'

Example: To cut 48 threads per Inch. Lcacksuew 2

Solution. 2 = 2 2X 2X 5 20x25x30
48 480 5x8x12 60x75x80

/or Screw-cutting on Lathes 55

(m) The Cutting of Left-hand Threads .

So fai, it has been implicitly taken for giantcd that only
though not often, that a left-hand thicad has to be cut. Foi
this pmpose, the leadsciew must lotate in an opposite diicction
to the lathe-spindle This is obtained by connecting up an
idle wheel at will, In double ttansmission, a fifth wheel (idle),
chosen at will, may also be mtioduccd

A number of lathes have been constiucted of late which
rend ei the connocting-up of an intermediate wheel un-
necessary. With these lathes, all that is icquned is to shift
the leveise-plate at the hcaclstock which reverses the move-
ment of the pinions which duvc the change-wheels, thus
causing these wheels and the leadsciew to lotate in an opposite
direction This is a decided impiovcment, as theie is not
much space to spare when five 01 six wheels are on the shear.
With a double compound tiain generally the larger numbei
are only small wheels, but with foui wheels, however, every
proportion is possible, so that the placing of a fifth wheel can
sometimes be vciy troublesome.

56 The Calt His twn c/ (Xv //<;?

CliAITlCK 111

AND HiHR iONMlUH 1ION,

TniSKK aic diffcient foiw. ol thread, a h-w of uhhlt aie
illustiated in Fi^s. 12 15

Fig. 12 shows the Ver thiead in its "rwtal fuuu, which is
conM.1 noted in different types, ,uul is inns! often met \\ith
Ki^. 13 illusti.itts the stjuau' <u ll.u (Ijit'.u!, the st-aion of
which is eithoi u S(|u,uc 01 a ii<;ht an;;lc, and wliit It is much
in use for luis^'i dismictois and >ai',n thuads. In IM*| 14,
the tiapexium tlucad is smi, Ihc Mrtin nt whu'h is a tiapc

r

I''J', 12, 1,3, 14, is.

aium, much in vogue {or the hvuKcrcws of lallu-s, the \\uun
being also a tiapeyfum thieati, FiV,, K; ,s the immd thu-ad,
ioimed by the iiiiej, section of snniYuvU's,

Voiy little need be said with ivft-nnut- t*t flu* last thm-
types, fur which it is inipossiMc to ,spak of any om- .sys.lt'in,
the form of thu section beitis; d pnnirnt tm u'n uinstanrrN
and detei mined by each individual at will

Different vaiiutieH, however, exist uf Ihe Vei* thrt-.ul.

(/;) Tj>/>?\ f

The type chiefly employed ts nnlainly the Whitworlh
system ; Mg, 16 shows llus constmction,

The depth of the Whit\uwlh threail t.s uqual tu u'<i,j of

for Strew-tzitmi> on Lather,

57

tin pitch, ihc sides of the thtead foi mini; an an^le of 55"
with top and bottom lounded tlnough \ of the line //,
drawn pctpemlicnlai fioin the- apex of the tnanide to its
base, tin* ladius of toundini', being equivalent to O'l (3 h

Not only is ihc sectional foiui oi the Wlutwoith tlnead
definitely fixed, but also the numbei of thieads pei inch foi

FK. 16

h 0,96 S

all dumeteis up to and including 6 inches, and this has been
fixed at from 20 JJ threads per inch.

The sectional loim i.s piecisely similai foi the finest as well
us the coarsest threads, and it is for this icason that the exact
dimensions and strength of the tlnead aie determined by the
simple determination of the outside diametei.

I >l.ltlUll 1 ,ll

Hi) (.torn

pwmch

i

<> as i

1 7^

20

;.

7-.!^ -A!

(.09

18

fl

c>*<52 , '&)

7-36

1 6

I\!

it'll 54

S'64

I'l

i

i."7o ' '39

9 <JX

13

ri

'5*^7 "S 1

I2't;2

II

l i

ID-US , '

15-74

It)

'{

^,"^2 '73

iH'S4

y

I

i<5'4 ' 'H4

21 '33

s

4

28 'W , '04

3 7

7

l|

it '75 i U 7

jsG'9-j

7

DitUtu'tet at Duimetei of

111

mm in. mm

>\$

34'(J2 l'i(> 2fJ'4<*>

Ij

38 'I I'^ij 32 '6S

*i

|1'37

i -37 35-28

Vj

4 ?J

j'4y V7 N

*>

Jo-ss

i 71 *nM3

3 i

57' 15

rtjj 49-02

55 '37

f0 4S

No of
Tliu.itK

per inch

4
4

31

70 '

3-03 6() Ho 3|

58 T/ie Cahulatitw <>/ Chatty II 7/rt-A

Table 1. mves the v.inous dimensions of tilt' Whitwoitl

A Whitwoilh thiead of in tain dimensions can also hi- ui
on a lonsuletabiy lai^ei otitsuU 1 diameter, the evut stu i n!' ( tl
of the tin cad beim> fixed by simph del CM mining; whirl
dimension of Hie VVIutuorth system is leqimed

T.ible I t'.ives not only the outside- diameter, but also tlu
cliametci at bottom oi llnead, so that the In ij'Jit nl the thiead
can be at lived at by subtia* tin, 1 * the lattei itom the iotmei,
and dividing the diffeienee by two

When cutting tlueads on the latin, whu h deviate in
ihumetei from this system, it is necv.saiy to know the depth

The depth of the tlnead < an also be airived at by a simple
calculation

Foi this piu pose, just look at Ft'i^. in. By diawim; a
peipendiculai horn the apex of the Uian;le t a ni'hl-antjled
tuangle is formed, the smallest am;le of \\huh is equal to
55"-r2 = 27" 30'.

Tuny. 27" 30' =O'^L Thciefbte, if the lunj| side of the
ught-any le = 1, then the short side ^ 0*5.^, and the base tif
the triangle of 55" ~ I '04

This base is, however, equal to h, i.e. the pitth.

Whence it follows that h : S = i : i '<v|, or o'*)ti : L

The ical depth of the tlnead is, however, only ;| li, So
that the latio between the depth oi the Urn-ad aiul the
pitch is equal to j h : S =r (o-yO x f t ) : i cs orj t | ; i Jj // tlm.s
equals O'6/i S,

If we take the outside tliametei 1), the diaitietei at the
bottom of the thiead 4 and the pitch S then, d ^ 1> **
2 x 0-64 S, or d I) i '28 S

in Table II., Is also according to the Whilwtnth sysluin, and in
fittings for gas, water, and steam by the AsstMUtitlon of
German Engineers, the Assocititioti of (icnnan Fltimber^ the
Association of the German Central Heating Industry, and
the Union of German Pipe Manufacturers,

for Scmv-tutluig on Lather

59

On the othct hand, in the autumn of 1898, an attempt was
made by ,i niuulu'i of mllnential associations oi Continental
en<>iiH'eis, assembled in confess at Zuiich, and including,
amongst nthets, the Association of Gennan Kn^meois, the

1 \III I II \VlIll\\URII! SlUlWlNl. Tllkl',Al>.

1' vti_tn il

in i nitu ! HI <mn in mm in nun in

h } 17! ^j u 71 ^( 8'^ 28 lA }8 ) l 882! 47 81 I 7<>S 44'

J j o 35 -qi8i} i->j hi u 41 "9 ill 1i'-7 ' u *' i S 1 i3 r 904 48

li ' 9 S' '<)S(t't(> f>7' '\$8iji4 9i 19 14 44 <|S- f H7 S 2 J 9} 49

S.'d

jj '15 H7J 9<>"-'*9'i 8 1 20''
,J 'it, os'f oj. lad 4l' '<).19 '(

14 2 50 8 U vi 7 59 f>i- 33 5f> 6 ^ u

1 1 2} s?* 1 ^- 3 5'^7 f'S 7- 2 "l7 i ()2 7^ n
H 33 03 5 '> 7^'-'^ S8a 73 =7

?, 2.-".v t uS<HO J 1-1)9727-87 14 .',' (ly-K
i 2tj'J t ^iKjhiV4 t ' 192 i. 'H ii 3 762

J47 247i M

4*5 ^8 SM 3 (lS
912 99 V'3 795

|3 75;* (J 5

-rjaj i 74\$

41-91 1-^338-95 H 4 101 (i 4339100-2 4223

4 1/32 i -028 41 '36 II

79 5 1 u

8S 5i II

9 f ' 39 i 1

[07" 26 II

AIII.K III. ~S. 1 Ti

ii.xm

1'itih

l)i imttu

at lUiHnill

1 Main

Ptlfh

1 ll.Ulll'U I

ui 1 hicail

))i mi

1'itrli

Duuuctii
nl Tluuwl

mm

mm

nun

iinu

null

mm.

nun

mm

nun

>

1

V7

20

2-5

if)' 75

48

S

41 ' 5

7

I

> 7

22

z'S

18 -75

52

5

45 '5

8

i *a^

^'37

24

^

2()'I

56

S'S

^ ^S

9

i j<;

7 37

37

?

-53 'J

60

S'S

57-85

10

rso

H'o<\$

v

3'S

25*4S

64

6

56 c)i>

H

i*9>

y>5

3^

3 S

28-4^

68

6

f)0*02

U*

i 7S

9*72

30

4

30*8

72

f>" 5

63 '55

M

2

IP4

39

4

33 '

76

(>'\$

67'5S

if)

2

I3'4

42

*l S

36-15

80

7

70-09

8

'S

M'75

45

4S

39 'S

6o

The Calculation of Change-Wheels

Swiss -Association of Machine-Tool Makcis, the Society
the Encomagemcnt of National Industries, etc., to replace
Whitwoith system, which is based on the English system
measurements, by a metnc thicad, and it was unanimously
decided to adopt the S I thread (" Systeme International ") '
as per Table III

Owing to the universal application of the Whitworth
especially of late years, this system is being more and
used on the Continent, especially by the Automobile
for thieads cut on the lathe.

FIG 17
The construction and form of the S I thread is mvera in.

o

Figs 17 and 18.

The apex is an angle of 6o n . The section is consequently
an equilateial triangle

Hence it follows that the peipendicuhu h, dropped from
the apex to the base, is equivalent to

The truncation equals | /t t so that the thiead has a
of 0-75 A, 01 o 6495 S.

for Screw-cutting on Lathes

61

Whilst the Whilwoith tin cad beais not only at the sides
but also at the bottom, the S I thiead, on the contiary, has
a play at the bottom of, at the most, ,',, //, equivalent to the
half tiuncation, the rounding of the thiead is equal to the

A/vv : \/v:.;n

\,

, D D,

\ A Y '
^/ ,* t

FIG 18

play, the radius of the loundmg in this case being j l () // The
rounding and play amount, as is generally accepted, to at
least ^ /{, Loewe stakes ,in aveiage foi this, and fixes the*
play and loundmg at ^,, h

The outside diametei of the male-si lew is thus smaller
than the diameter at bottom of the thread in the nut, and

FKJ, 19,

vice versA, the diametei at bottom of the thread of the male-
screw is smallei than the outside diameter of tin cad in the nut.

62 The Calculation of Change- Whech

If we take the play a, then the actual depth of the tin cad
of both male-sciew and nut equals o 75 // + u If we fix the
play at its maximum, equals T \, k, then the height equals
0-0625 /; + 0-75/2 = 0-8125 /*, 01 o 703625 S = ~0'7S.

The Lowenheiz thread (Table IV) is in j>onei,il use up to

dim

Pitch

Diimetei
at Bottom

Diam

Pitch

Dmuctu
it llolttiin

Diiin 1'itih

Ui unitu
it Bottom

Of IllK.ul

mm

mm

mm

mm

mm

mm

nuu mill

HUH

I

o 25

o 625

2 6

O'

'45

I

925

S'S y

4

15

I 2

o 25

o 825

3

O'

'5

2

25

6 i

4

S

I 4

o 3

o 95

3 5

o

6

2

6

7 i i

"5

35

I 7

o 35

i 175

4

o

7

2

95

8 J 2

6

'2

2

o 4

1-4

4 5

o

75

3

375

9 i *

7

<>5

2 3

o 4

i 7

5

o

8

3'

8

10 i 4

7

9

a diametei of IO mm foi instiuments of eveiy description,
especially in Gei many and Swit/eiland, and in scicw works,
the sciews are almost exclusively made by this system

The construction of the Lowenhci/ thie.ul is shown in
Fig 19. The apex is 53 8'

This angle icsults from h ~ S. The tin cad is titmcated
flat on the outside diameter and at bottom with a J truncation,
so that the ical depth of the thread is = 0*75 //.

stiucted as pei Fig 20

for Strew-cutting on Lathe*,

TAUI v, V - Sw i FRS THREAD.

Nunilii r >{

I'll! < U.I'

)ll 1 UK ll

Nmubtrof
I'liu ul
pen mi h

1,1 Mini ter

Ntuuhii ol
Hiu id,
pu inch

.JO

24

20
IS
16

M
M
ta

JX
1C)

5

S

-ll
4
4
31

31
3i

4
4l

5

Si

SI

si

6

3}
3l
3
3

The apex is t in an^le of 60, so thai the peipcndiculai f {
(hopped fu)in the apex to the base, is again = o*866S, The
tlueacl is Hat-faced at bottom and on the top with J trunca-
tion, consc(|tiently

t = ,"] fj and 0'7S X 0'866 = O"6495 S.

has, however, no play and is divided accoiding to English
measuiements.

Although largely displaced by the Sellers thread, theshaip
V thiCiid still exists and is used in America, (See Table VI.)
The section is an equilateral triangle not truncated.

6 4

The Calculation of Change-Whccl\

The B A. S (Bntish Association Standaul), as per '1 able
VII, is an English tluead much used in Kn^land foi sne\\sof
small diamctci, especially foi electiu fittings. Tin 1 - a|H \ t
Fig 21, is an angle of 47 \ n The thiead is tninratt'd, and

' VI SUAKl 1 V I'lIKI AD

Number of Number ot Niiuiln i of j Niimlii i nj

Dnnit-tcr rineads Dmiitlc.1 I liu uU Diumui Iliiivid hiuiiilii J'iin ul ,

ptr mi h

pi.r UK h

inch

inch

mi h

i

20

Hi

10

n 1

ia

IS

I

o

i/t

,!

16

1 fi
l ii

9

>

I 7 i\

14

I

,S

* H

i

12

Ik

7

2 !

A

12

1}

7

2{{

8

II

I ft'

6

2\

3 1

II

ii

(>

^

i

IO

1 5

"5

j)

top and bottom aie rounded, leaving the depth of the
equal to o 6 S

In addition to the foi cooing, the Delisle, Sauvnje, Acmt 1 ,
and Thuiy systems aie to be met with.

The total number of thiead systems exceeds fifty, hui only
the seven most used have been treated of ht'ic,

Number

ta

l r m etei> 6 5'347 4'i 3'6 4 va 2 H 2'S '. i-Q 1-7 r 3 i
^ i 0-90 810 730-660 Spo'S^o^Ko'o'io-.-

Jot" Xi

on Lathes,

((} Si i ew-c lifting Tools.

A tool used ioi sucw-cutlino m iist fust and foiemost be
peifet tly tine. It is not to he looked upon as an oidtnaiy
tool, noi may it be mound on a stone which does- not mn line

When tuttim; deep threads, whethei they be V 01 squaie,
it is always advisable to use sepaiute tools loi toughing and
hmshim,

The cutting an{le must be about 70, whilst the tool must
not !r pointed 01 somi-t nculai, but flattened at the cclqc
(\ f li\\ n 2 and ?3), as othenvise the angle will not be true, and,

.it the same tune, it will be impossible to nrind the tool
accurately The tool must not only stand on its edge in the
alible B, IMJ,' 23, but the sides A A must also have cleaiance,
The atisde in which the thread lies on the work has also to be
lakvn into etmsidetation, and the line AB, Fig, 24, must tun

.&

Fie.. 2<j, Fir., 26.

at the same angle. Suppose that a I in. pitch has tu be cut
on n diameter of 2 in. Then, imagine C D in Fi^, 26, to be the
angle a I which the thread lies on the woik, the line AJBof

F

66

The Calculation of Change II '/r<-A

the tool, Fig. 25, must thus run patallel to the line (' 1) in
Fig 26 This is still moic evident in the i ase of sqnaie
thi^cads with a coa ise pitch, Fig ?7 In this i ase, tlu <kuan<e

on the sides of the tool must be diOcient, The diametei of
the tin cad on the top, as ,ilso the angle of the tluead thete
is indicated in Fig. 28, that at bottom of the tluead in
.^ Fig 29, a and b being the ehrumferenee,,

t and c the pitch, which is tlu* same for both,
and thcic aie consequently two angles*
The hypotenuses if/ and <* show the un^lc of
the tin cad til top and bottom. If the
cleaianceol the tool is tont'ct on the top.it
will be mcoueet when at the bottom. The
hteepei the pitch, the moie noticeable this
will be. The tool must have more elearanr e
on the right-baud side for bottom than ut
the top, but less on the left-hand side,
The tool must consequently be ground in
such a manner that the tight hand side will
have enough elearauu' at bottom of the
thread, whilst the eleanmre foi the loft-
hand side must conr.iu with the angle ul
the top, that is to say, foi a light-hand thread, as in Fig. i*7 ; for
will exist in icgard to angles. The tool must menrdingly be.

It C-i

FIGS. 28 AND 29,

pound as indicated in Pu,> 27, A R being- the slope of the
iKjht-hand side ot the tool, A C on the left-hand side. The
iippi-i cutting smfart' of the tool must um squat e on the line
A l>. When <uttim> an inside linht-lund thiead, ey;ciythinj>
is i excised, wh.it is li^ht-hamled becoming left.

Foi a Vee tht cad, the tool must be <round in accoulance
\\ith the aU",lc- of the system ol the Ihioatl, ft need scniccly
be said that this must not be left only to eye 01 the lough
estimate of the opeiatot, A tauge should be piovided, as

V'W'I'r'i!^

pel 1'V, 30, ;ivnv', the pierist* un^le. And yet, notwithstand-
ing that it is lai nou k diHieult foi a woikman to judge an
tun;/! 1 \uth llu e)t- thun to ^uess i ceitain /etigt/i, and no one
would eun think of peimittin^ an opeiator to estimate a
u'lt.un length without usinu his rule, it is- an exception when
the opnatot K ptovided with n suitable an'le gauge.

// /\ nttt'i'ty iw/xnA't/'fc that a thread can be trite when llie
tfuttttH JIM juried //!<' rW/i' J the ttwl with his naked eye.

This {juuuc
purpose 'kvrn

FIG,

a .second, and not less important,
the tool be gnound to the precise

It is htill possible to out a wrofl^ thread, for the tool
must be M pkeed in the holder that an imaginary line diawn
pinpcndteularly tnn the apex of the triangle to the imaginary
ImstMUUht alho fall perpendicularly on the side of the cylinder
on which the thread is to be cut, Not having this gaugc,

I? 2

68 The Calculation of Change-Wheels

the opeiator judges with his eye the position m
thinks the tool should be placed But the most
woikman can make a mistake, it is not possible ^f
to be ti nc if the tool has been placed with the
in the position which might icasonably be
collect, and this is afteiwaicls checked with an

FIG 32

it will almost invaiiably be found that the position r
The icason is that the two lines forming the ang>-lc .
vciy shoit in piopoition to the othei lines of the t:o<
being consequently deceived

In Fig. 31, at A, Is shown the manner of gauj^inj
to which a lathe centic should be tinned, at B, tl

FIG. 33

which a screw thi cad- cutting tool should be
C, the concctness of the angle of a scicw thieacl -<\
In Fig 32, the shaft with a sciew thiead is ^vip
held between the ccntics of a lathe. By apply In
as shown at D or E, the tin cad tool can be set zxt:

for Sircw-cnttwg on Lather, 69

!<> the sh.ilt, and then fastened in place by the boltb m the
tool post, tlu-U'by avoiding nnpeifect 01 leaning thieacls

' n I'" 1 ", 33* <-t V and G, the mannei of setting the tool for
< lilting internal tin cads is illustrated

As pievumsly stated, it is always advisable to begin
cutting a IhuMcl th.it has anything like a deep curve with a
loughing tool which us at a cutting point and which need not
be {Ljiound piecisely to the angle.

The thiead should afteiwaitls be gone ovei with a
finishing- tool When engaged in cutting shallow threads, the
tool can cut on both sides at the same time, and it can be
put exactly on the duection of the shaft. With deeper
tin cads, i.e. quick pitches, this is no longei possible. Cutting
with both bides of the tool at the same time causes it to snap,
the thiead is tough, and veiy often it is impossible to continue
working; the tool should, theiefoie, woik but one side at a
time, should fiequently be set slightly m a paiallel duection
to that of the shaft, and cliiectly theie is any play between
the tool and the thread, it must again be set squat e on the
direction of the shaft. Each time that the tool has gone
completely ovei the thread, it should be withdrawn and again
set in the oiignul position at the commencement, though
increased with the amount cut at one passage

For this piupose a gi actuated collar is provided to the
feed screw by means of which the traverse movement can be
read, and by which the tool can be set in the exact position

every time.

"The opciator foi racily got out of the difficulty by maiking
the position of the screw spindle with a piece of chalk,

On lathes of up-to-date consti notion, the giaduated collar
is now always to be found on the sciew spindle.

A very practical consti uction is shown in Fig. 34-
Advantage is hue taken ol the movement of the two half-
nuts when opening and closing, to withdiaw the cutting tool
irom the cuive, and mis vcrsd, back again to the exact

The CaknlafKni of C/ian^e-ll

position, so that instead of having to cany out vaiious opeia
tions at the end of the thtead, a simple movement of a
handle is till that is icquiied.

The constiuction is as follows Ovn tin* (\\o half mils
which move unclci the carnage in the .same du tit ion as tin
cioss-slide, and aie opened and. closed by a double n;;hl- aiul
left-hand sciew, is placed a Pi-shaped slide lived mi knobs of
the upper poition of the half-nuts, 'I he su< w spindle ol the
cioss-slidc fits in the uppci poition of this slide on the one
side by a turned up ed^e, and on the othei by huk-nuts
The scicw spindle must consequently follow the movement o!
the slide Holes aie dulled ii ( idU thtoii",h the ptojetthu 1 ,

/

v r , ,, * ' I I "\

j- -^,J,ir""r J^' ",, * j f ^ r U ^)

paits of the half-nuts, and the slide, A steel pin lit-, t fnsrly
into these holes. Oblong holes, in which the pin has play, air
boied in the carnage for same.

Before beginning to cut internal or evtenul llm-suK, tlu*
pm is set in the foiemost 01 hindmost nut, so that the halt nut
thiough which the pin is placed Ls coupled with the slide In
which the scicw spindle fits, and umseqiiemly they imtsl
follow the movement of the half-nut in que.stion lont-ther
with the cross-slide and tool. It is wotked as Jollmvs: As
soon as the tool has anived at the end ol tin- threat!, the
half-nuts of the Icacl-scicw are opened and by this means the
tool is withdiawn fiom the thread The caniagu j then

liftin on

71

n tinned l>\ hand hy means of the pinum, the tool set so
imuh faithet in with the sue\v spindle as it is desiiod to cut
drepej, and tin- half-nuts aie closed again This causes the
tool to lesnnie Us oiigmal position, only cutting the uiatcnal
st much deeper as it has been set Luther in by hand. If no

thtead is lo 1> ( nit, the connection between the slide and hall-
nut is Inoken by withdrawing the locking-pin, and the slide is
coupled to the uoss-shde by insetting the pin in the hole
boied tlmniyh both slides

When screw cutting, this anangeinent ies.ults in a decided

I

saving of tune, l>esiilcs preventing the possibility of mistakes
arising from inserting the tool eithei too far or not far
enough In,

There should be an outlet for the tool at the end of the

72 The Calculation o/ C/MH^C l\ f Jn.cL\

thiead If the diametei is sufficiently lai;>e to pet mil of it,
an entiie circuhu gioove should be tinned, Ki<> ^5 If, ioj
some reason 01 othci, a citcul.u gtoove is not possible, a
suitable outlet, as pel Figs 36 39, must be dulled foi vee ot
squaie thieads Bcfoic commencing cutlin<j ( , the tool should
be so fixed that it will antvc just at these holes.

It was foimeily the custom to tetuin. the uuna^e when
the tool had gone over the tlne<id, by tevoishi", the movu-
rhent of the lathe But with the ptesenl-day eonsiuu tion
of the lathe, by which it is possible to letnin the carnage
quickly by hand by means of the handle, the h, Ul-nuts ate
opened and the caniage leturned by hand. If the thiead bcttu;
cut is of the same pitch as, 01 an aliquot pait of the pilch of
the Icadsciew, the half-nuts can be chopped into engagement
at any point of the Icadsciew without any difficulty, the tool
always ictuimng to its pi ease position in the thiead This
is, however, not so when the nmnbei of thiead s pet inch aie
uneven 01 bioken, and othei means must be adopted to ensnie
the tool ictuintng to its precise position in the gioove. Con-
sequently, when slatting to cut the Ihtc.ul, a slop, ot mm king
line, is placed on the bed, the half-nut closed and a ehalk line
diawn on top of the leadsctew, and anothu i hulk line nt the
fiont side of the chuck-plate. When the tool bus none oyet
the thiead and the can tug c has been teUnned by hand as Jar
as the stop or the line, the head spindle is turned lound till
both chalk lines are again in then original position, the nuts
closed, and the tool is once moie m its pieuse place in the
path which has, just been cut,

This compaiatively troublesome and punnUve manner of
working is done away with, if the c;iniaf;e is piovklcd with a
thiead indicator as shown in I ( "i^ 40,

The following is the principle of this uttat hment ; A stwvll
worm-wheel runs on the leadBeie\\ and by moans ol a pinion
gcunng, causes an indicator to move on a (iicular indox-
plate. All that in now nuees-saiy is in iuli* the position of
the indicator at the starting point, allot which, tlu: half. nuts
can be closed, mid the tool will come pieeisely in the path
each time the indicator resumes its original position,

for Screw-cutting on Lather 73

(?) The Cutting of Double or Multiple Threaded Screws

The cutting of double 01 multiple threaded sciews causes
<i ^<>od deal of tioublc, as, in addition to cxeicisiny ouhnaiy
uue that the thicud cut is tine, anothct most impoitant point
has to be taken into consideration, viz that the setting of the
tool is also exactly equidistant The manner of working is
smiilai to that foi a single thiead, but caie should be taken as
fat as possible that when cutting a double thiead the spindle
wheel is divisible by two, and foi a ticble thiead by three

After the fust incision has been made to the lequned
depth, the tool must be shifted exactly to the centie between

Fltr 40

two threads for a double, and to one-thud of the intei mediate
space for, a treble thiead The distance the tool is to be
shifted should, however, nevei be mcasuied off, as this can
novel be exact, but must be obtained by mechanical means,
either by turning the woik-piece while the Icadscicw is
slationaiy, or by turning the leadsciew while the work-piece
remains stationaiy, If a double thiead has to be cut, one
of the teeth of the spindle wheel coming between two teeth is
maiked with chalk, as also the two teeth which the tooth in
question engages., After this the spindle wheel is bisected and
this tooth is also chalked ; the spindle wheel is then icleased

74 The Calculation of Chaugc-Wltceh

from the wheel it engages, the spindle is given half a turn by
hand, so that the opposite tooth comes between the two
maiked teeth, and the two wheels rue once moie engaged
If the spindle wheel is not divisible by two, then this must be
found on the wheel on the leadsciew, but the pitch of the
thiead to be cut must in this case be taken into consideiation
3 inches is to be cut on a lathe with a leadsciew of 2 thicddfc.
per inch.

The fi action is -i-= 3 = 7 *> .
\ ^ 5

The spindle wheel is, howevei, not divisible- by 2, and at.
the factor 3, which is indivisible by 2, will invanably be found
in that wheel, 4 wheels aie used so that the factot 3 can be
placed in the intei mediate wheel

75 _ 100 X 60
50 ~ 50 x Bo

If there is any reason, for instance, with heavy lathes not to
turn the spindle but to shift the carnage by turning the lead-
screw, this is accomplished as follows foi the above example

Pitch = | in The carnage must thus be shifted ,"/ -f- 2 =
| in , the leadsciew has a pitch of ^ in., and so must make
3 1 = 1 revolution, the wheel of the leadsciew has So
teeth, and consequently 80X7? 60 teeth must be moved

If the same pitch is to be cut on this lathe but foi a thtee-

thread, then the first-mentioned wheels, ?5 arc the best to

SO

use , the wheel with 75 teeth can be divided into thiee, and
25 teeth turned each time.

If it is desired to move the caniuge, this must be moved
T-T- 3 = i i"-, the leadsciew make 4 i evolution, and the wheel
with 50 teeth be moved 50 X J => 25 teeth.

For example, To cut a pitch of ij in. Double thieaded

Solution IL = J 5 = I0 x 6u

I 8 40 X 80 "

jor Sit'ciu-cut/iug on Lathes. 75

Poi ,i double Ihieaded sciew, the spindle wheel is divisible
by \

\\ -i- -3 = J~; in Thi.' lead sci ew must thus make { (j ~- l
} [\ revolution,

|[| x Hi) = 75, The wheel on the leadseicw must thus be
moved 75 teeth,

K \*amphv~ To cut 6 thieatls per 15 in, thiee-thieadcd

c i , If' 15X2 75 X 80

Solution ' JUi_=: > = ' J

\ () 30 x <\o

Poi a thiee-thie.ul, the spindle wheel can be divided into
3 X 25 teeth

The Cciniagc must be shifted \, 5 3 = il m ' so thal thc
leuclscicw must make ,] -f- J = V' i evolutions

The wheel with 30 teeth is placed on the leadscicw, and
30 x \l } - 50 teeth ate moved = 50 -r 30 = J i evolution and
20 locth.

(/) 77/6 1 Cutting of vet y Coarse Thread.

When cutting coarse tin cad, a difficulty may possibly
occur which will lequire cuicful consideration When the
tin cad to be cut is considerably codiser than that of the lead-
set ew, the movement of the leadsciew must be appreciably
quickened, There is, however, a limit to this, and that is the
H-sistante olfeied by the teeth of the goat -wheels If the
pitch is too coarse, these will bieak off The extent to which
the pitch may be mutased depends, rmtmally, entnely on
the stien^th of the wheels supplied with the lathe, Generally
speaking the pitch may .safely be a four-fold of the leadserew,
anything exceeding this being attended with considerable
danger and the off-chance of the teeth breaking.

hi order to permit thread to be cut which is many times
toaiser than that of the leadscrew, a gearing can be attached
to the fast headstoek, as illustiated in Fig. 2.

The wheel 15 can be set in connection with the small geai-
whuel of the double back gearing. If then the lathe mns with

7 6 The Calculation "/ Change- 1 ! 7/< , A

double back gea,, the ..ilu, "I -P-l l-euveen UH ,..,- Pulley
and lathe spm.llc w,ll lx- I K, 'll IK-IWC-,,. .lu- .-1 . , ... ul
the cone-pulley pm.on 2 ,, ,u,,l the u'h,, 1 ,u w,l U,,,,,,,!, U
4 .evolutions l<> i of Ilu- sp,,,.IU- Inllu-.-.s,.,,!., ll,,,...,l
vvh.ch ,s foiu times auisci th..n the l,.,Kuew, Iheu ,, .1 ,.it,..

of I ; i between the t-han^t; \vlnvis, whil ,( tui ,i ihir.ui i-s^JH
times as coaise, Lheie is only a ratio oi i; TaUm", ,f uii\i*
th.it the teeth die stump, etioii-'Ji UH a talus ut I ; ,|, aiul that
the leadsctew has i in pitiii, then u \ x.j X | K HI, pit'h
may easily be cut in this l

lor Si moulting on Lathe\.

77

One of the in \\i-st designs fot sc rew~cuttm^ is that of the
litMhl, v-Muth.n Msicm, \\hieh, ), y HUMUS O f attain of ocais
p!u<vd umlei ,md ,U the side oi the hcrulslock, icndeis it
l" IN '' !blr to (Ul 'i nuinbei o| thio.icls o( thfToicnl pitches

)s ~v*;,'v.v {'"' ^'^^(tWMJSSff MS f /////\$/>\$,

, i ' ' l V '\NN\\V. '11 * v-if^jMc "V 1 i *W

H 'TfT ... ! ' ^ ' ' i! i! ^ !-i

irn

FfC 42

with* nit the iHTt'ssity of fixing different ch<mge~wheels.
("halloo wheels as they have up till now been imdei stood
JH t'nnnv{}tii with the htthe, luive been entirely supuiHcdcd
On u Ltlht 1 piuvided with the llendey-Norton system, it is no
lon;;<M ju'tt'ssary to fix up or lake off change-wheels, the
various whooK being .simply and solely geared tip in the space

78 The Calculation of Change /77/6rA

formed between the spindle and the leadsciew by the shifting
of handles The calculation of change-wheels is consequently
a thing of the past

But, in this woik which tieats of the whole question of
screw-cutting in an nbndi>ed foim, a descitption of this
system, which will ceitamly come moie and inoie to the fmnt
in the stiu<lcfoi economical tools, and has ahead} been vciy
largely adopted, must not be missing

" 4 " , VL *> m^'T^s
.- /A X^m^i^

\ ', ;,'?:' . \ ^ **

Aruuigemenl o{ wheels m a L<xl|;i> anil Shipli y hulu, the dutt

On a lathe of this description, scicw-t uttin^ has Ix-cn
i educed to its simplest possible foim, A drver \vtu*Kaitn
may, it is tine, be quite capable of e,duil,itin } ", the \vlu k cis
leqmred to cut a certain tin cud quirkly, and cun possibly
reckon it out m his head, but even so, llu- utlual fixing up
of the wheels seriously interferes with the strady pu>^rcs of

for Screw-cutting on Lathes. 79

the woik, whilst the difficulty is at once doubled whenevci
turning, chilling, and thiead-cutting have to be pcifoimed
periodically, as, with so many lathes, the attendant chcum-
stanccs ,ue such that it cannot be aridiigccl foi all at the
same time

The lathes undci discussion ate constituted in such a
manner that a gieat vancty of threads can be cut without
icquuing the fixing up 01 taking off of a single wheel

In the eailici constiuctions of this type of lathe, theic was
mvanably one gieat drawback, viz. that the numbei of
pitches which could be cut was comparatively small (10-12
pitches), but this numbei has now been extended to fiom
40 to 44 diffeient pitches.

The foicgomg illustiation (Fig. 43) shows the complete
anangemcnt of the wheels

This gives a cleai view of the bed, the fast Deadstock
having been icmovcd foi the piuposc

The anangement of the wheels consists of two sepaiate
groups of wheels The fust gioup (9-11 wheels) is placed
undei the headstock, the second being in a closed box
attached to one side of the lathe

The action peifoimed by a wotkman in geanng up the
wheels foi the cutting of diffeient pitches is cxtiemely simple,
so that aftci a biicf explanation it is sufficiently cleai even to
a novice, tmd it can be executed so quickly that not more
than fiom 10-20 seconds aie icquiicd to change the wheels foi
anothci pitch than that foi which they weie geared up.

An index plate is affixed to the gear-box, which is given
on page Bo In its exact si/e.

A handle with pointer is placed undci the plate. This
pointei can be moved over the cntue length of the index plate
and set in the middle of cithci of the foui divisions of the
plate. This handle is connected with the wheel indicated in
Fig. 41, by the number 862, winch accoidingly moves this
wheel with it , whilst nuclei the holes in the headstock the
numbers I 8 or i~ii appeal, accoidmg to the dimensions of
the lathe.

If, lor example, it is leqmred to cut \$2 thieads pei inch

8o

The Calculation oj Changc-irhcch

Thds

Knob

Thds.

Knob

I lid',

'I lid

KlHl

18

19

20

22

23
24

26
28

3
32

3
4
5
6

7
8

9
10
ii

9
<)1

10

ii
ni

IS

13
M
IS
1 6

3
4
5
6

7
8
<)
i<>
ir

I 1
J, 1

S

s 1

Si
6

('
7
74
16

3
I
S
(.

7
S

It)
1 i

(I

7

S

<)

lo

I J

80 to 40

40t() 20

Pi A I I?

It) to

the pointci is placed by means of the handle in the middle of
that division in which the numbei in question appeals mulct
the letteis Thds. (Tlneatls), in this paitieulai ease, in lh"
second division on the light hand '-ide. On the same line on
which the number 5] appears, the fignte f t 1 ? to be found.
The handle on the head stock is now plated m the hole above
the figure 6, and the wheels me than <>eaied up foi euttitif*
index plate, the ptocedme is identical The topmost handle
957 ls pl acc d m the highest 01 lowest position, uivoidhi^ as it
is desired to cut left or light hand thread

We will now pioceed to give a detailed description of the
constmction of this gcanng.

Wheel 968 (sec Fig. /|2) is fast on the lathe, spindle and
engages wheel 922 (Fig, 41) whunevet right -hand thread is to
becut In this case wheel 933 is idle. Kot , i left-hand threat!,
wheel 968 engages 923, and wheel <).L' i.s caused to total e by
wheel 923, so that the direction of movement is just the
rcveise to that in the fust case. Both wheels run loose on
studs fastened in plate 920, and arc- shifted by the middle

for Screw-tutting on Lathe*,. 81

handle Wheel 922 engages wheel 955 which Is fixed on
shalt 992, which is consequently biought into motion, This
same shalt ^52 nnpaits motion to wheel 959, which, by means
of a key way, can be moved in a tiansvcise diicction by the
handle undei the fast headstock Wheel 959 engages 961,
which can be geaied up, by means of the handle ahcady
icieued to, with all the diiletent wheels 651-659 mulei the
fast headstoek, which wheels aie all fixed on shaft 662 , wheel
961 consequently impailhig motion to the shaft. Wheels 666
and 067 ate also kej ed to shaft 662. Wheel 862 (Fig 41)
movable by a keyw.iy, is mounted on the leadscrew. Con-
sequently the motion of blufl 662, to 'which the gem-wheels
ate keyed, is ttansmitted to wheel 862 by one of the wheels
606 01 (167, vid two sets of double wheels 905 and 906, both of
which setsaie identical

This tiani of geais can be seen in the detailed drawing*
Ktj- /jl, to the lelt ol the side view of the fast headstock. Jt
should be noted that wheels 905 and 906 aie coupled, but that
each set is independent of the othei, and can consequently
lutate at diffeient speeds , this is, moieovei, appaient with the
whole turn of geais, seeing that, whilst wheels 666 and 667
also coupled, and each engages one of the sets 905 and 906,
the latter obtain vaiious speeds. This tiam of geais gives
four diffeient speeds between shaft 662 and the leadsaew

Wheel 606 engages 905 and 906 on the right Wheel 667
engages 005 and 906 on the left,

By moving wheel 862 on the leadsciew (this wheel is
also to be seen in the illustration, Fig. 43), and by changing
handle ydj, which tuins on shaft 662 and to which at the
same time the two sots of wheels 905 and 906 are keyed,
wheel Kfo can be placed in lorn different positions, i, 2, 3,
and 4, (See detailed dt awing, Fig. 41.)

Wheel {>(; =- 906 and wheel 666 = 905 =s 862. The
proportion of 007 to 906 .= r I, of 66(>to 906 =2 i, so that
If wheel %j engages 905 on the light, the speed of shaft 662

* ^ V* ^

i,s doubled, seeing that ** 2,
** i X2

If wheel B63 engages 906 tm the right, the motion ol the

G

The GittM/tithw of C/Miigt- //7v'A

shaft is tiansmitted without nn)' vanation, am I ulu-t i <,<< n
the light simply serves as an idle wheel, li ,H* > t n-; ,t",e<, 'in;,
to the left, tlieie is a double i eduction in sped, U ,Sn* < n^a-M
906 on the left, the diminution is lour tunes is 'iieat. <*tm
scquently, if the handle on the fast headsttu k is set t <j nut"
No 9:

With the pmnti'i in uihmw J, ^ iiUh<', pt-t n It
*'. '/'
). *4

4, 2K

will tn nt

In this manner, with n wheels on shaft fh\ 4 j tltllei
pitches can be cut.

The swing plate of the f.ist headstoek IN hutlun * tun
structed that, by netting up oius wheel, the spt'ttdu! tlt< lt%w!-
scicw can once more be doubled, or by removing fin* ?..wu
wheel, it can be reduced to half as slow ugahi, so that all flu
threads appearing on the index table can now be ujl, with
double or half the number per inch, The reseive hle tn the
swing plate can be cleaily .seen in 1%, 41, ulM* to i,>,*j,

In the foic^oing illnstraticniH, Fig, 44 glvc;H the ebljit-
lion foi fine threads, Kig, 45 for coarse threads whilst W|t, *|6
shows the position of the wheel 955,

for St rew~i ntting on Lathes.

83

Tin- iisu.U ;e.utm Is: Wheel 968 engages 922, and 922
ur;.tn's u^, the wheel on shaft 9152. For fine thieads, 968
\ U'si'tes OJ~, and c> >2 engages o?3, consequently 923 engages
t> {, n'luch IN ,i double whet I with 9?\$, the proportion between
them being i 2, Finally, 923 engages 955 Wheel 955
d*H% nut engage <)2\ hut is moved a wheel's width to one side
(See Fig, 4] 6)

MfH^iAft TffltCtOS Of /NOKK

Frit. 46.

Km rtmisi^ threads, qf>8 engages 922, and 92
i.tiu't|iwntiy ^1 imij.mt's 923 and 923 engages 9^5. Fiom
,i t',m*lut ionsith-ration tf these two comb!natkm& foi fine and
MMi.r tlsie.nN, it will he seen that wheels 924 and 925 on
tiu* OIH- suit;, atul wheel 923 on the other side, are mutually
iftfti*h.ini',*Hi foi the two cases.

Sfai It !WH ontyhocn multiplexor fractions of an inch,
in both, which nmia !>o cut in thin manner. Should it, how-
rvu- ht Ht'WN-ary to deviate boredom for any special pitch,
tlwi thrr,tits than ihoscof the KnjUHli system can also be
,-irt by 4 u-ruin proportion between the two wheels 924 and

0-5. 3992

/i

```